In this thesis, three crucial questions arising in multi-objective optimization are investigated. First, the existence of properly efficient solutions via scalarization tools is studied. A basic theorem credited to Benson is extended from the convex case to the general case. Some further scalarization techniques are also discussed. The second part of the thesis is devoted to robustness. Various notions from the literature are briefly reviewed. Afterwards, a norm-based definition given by Georgiev, Luc and Pardalos is generalized to nonlinear multi-objective optimization. Necessary and sufficient conditions for robust solutions under appropriate assumptions are given.

Relationships between new robustness notion and some known ones are highlighted.

Two kinds of modifications in the objective functions are dealt with and relationships between the weak/proper/robust efficient solutions of the problems, before and after the perturbation, are established. Finally, we discuss the sensitivity analysis and stability in parametrized multi-objective optimization. Strict semi-differentiability of set-valued mappings of feasible sets and feasible values is proved under appropriate assumptions. Furthermore, some sufficient conditions for semi-differentiability of efficient sets and efficient values are presented. Finally, pseudo-Lipschitz continuity of aforementioned set-valued mappings is investigated. iii Résumé Cette thèse porte sur trois questions qui se posent en optimisation multi-objectif. Dans un premier temps, nous étudions l'existence de solutions efficaces via des techniques de scalarisation. On étend le théorème de Benson du cas convexe à un cas général. De plus, nous examinons d'autres techniques de scalarisation. Dans un second temps, nous abordons la question de robustesse. Nous examinons les concepts proposés dans la littérature sur le sujet. On étend au cas d'optimisation multi-objectif non-linéaire la définition de Georgiev et ses collaborateurs. Quelques conditions nécessaires et suffisantes pour obtenir une solution robuste moyennant des hypothèses appropriées sont données. Les relations entre cette notion de robustesse et certaines définitions mentionnées sont mises en évidence. Deux types de modifications des fonctions objectif sont traités et les relations entre les solutions faibles/propres/ robustes efficaces sont établies. Le dernier chapitre est consacré à l'analyse de sensibilité et de stabilité en optimisation multi-objectif paramétrée. On montre sous des conditions faibles que la multi-application de l'ensemble des solutions réalisables et des valeurs réalisables sont strictement semi-différentiables. On donne quelques conditions suffisantes pour la semi-différentiabilité de l'ensemble efficace et des valeurs efficaces. De plus, nous étudions la pseudo-Lipschitz continuité des multi-applications ci dessus citées.

Preface

Decision making is as old as mankind on earth and everyone somehow faces this issue everyday. In this problem, Decision Maker (DM) has some alternatives among which he/she wishes to choose one with the most beneficial achievement. In most cases, DM is faced with conflicting objectives that make the choice of best alternative complicated. To clarify this let us give an example. Consider a person who is going to buy a car and their criteria are fuel consumption and power. It goes without saying that a car with high power burns more gas. There is no single alternative that simultaneously satisfies the two criteria. How can DM make a choice?

As in a decision making problem, one is supposed to select merely one alternative; he/she has to trade-off among objectives and opts for an alternative. However, it would be profitable to distinguish alternatives (feasible set) which can be candidates.

Edgeworth 1 is among the first to address this issue. He considered multicriteria economic decision making and regarded an alternative as an eligible candidate if there exists no feasible solution for which an improvement in one objective does not lead to deteriorating at least one of the remaining objectives. Vilfredo Pareto 2 in his famous book 3 studied this problem and called these candidates as efficient points.

Nowadays, in the literature, decision making problems are classified and treated according to the nature of the feasible set. If the feasible set is finite and given explicitly in the beginning of the solution process, it is handled by multicriteria decision analysis (MCDA); otherwise, it is treated by multi-objective optimization. However, both paradigms could be applied to make a successful decision. MCDA is out of scope 1 Philosopher and political economist, 1845-1926 2 Engineer, sociologist, economist, political scientist, and philosopher, 1848-1923 3 Manuale di Economia Politica, Societa Editrice Libraria, Milano, Italy, 1906 of our work here and we refer the reader to the book [START_REF] Evangelos | Multi-criteria decision making methods: A comparative study[END_REF] where special techniques like AHP and TOPSIS are introduced to handle these problems.

Multi-objective optimization is a part of mathematical programming dealing with the minimization of some conflicting objective functions over a feasible set. Multiobjective optimization problems occur in many fields including economics, management, multicriteria design optimization, water resource planning, medicine, etc. Up to now a great deal of work has been devoted to multi-objective optimization, from both theoretical and numerical aspects.

Numerous approaches have been proposed to solve multi-objective optimization problems including scalarization techniques, interactive methods and so on. Scalarization is a traditional approach to solve multi-objective optimization problems. By scalarization methods, one solves a single objective optimization problem corresponding to a given multi-objective optimization problem whose optimal solutions can be efficient. Strictly speaking, a single objective optimization problem is formulated related to the multi-objective optimization via a real-valued scalarizing function, typically being a function of the objective functions, auxiliary scalar or vector variables, and/or scalar or vector parameters. In addition, the feasible set of the new problem may be restricted by new constraint functions associated with the objective functions of the multi-objective optimization problem and/or new variables are introduced [START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Eichfelder | Adaptive Scalarization Methods in Multiobjective Optimization[END_REF].

Scalarization methods have been employed like an engine in interactive methods and approaches for estimating efficient sets. Moreover, heuristic methods such as genetic algorithm, ant colony method, etc., have been widely developed for solving multi-objective optimization problems in recent decades [START_REF] Deb | Multi-Objective Optimization Using Evolutionary Algorithms[END_REF]. One of the solution concepts which plays an important role in multi-objective optimization, from both theoretical and practical points of view, is the proper efficiency notion. This concept was first introduced by Geoffrion [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF] to eliminate the efficient solutions which cannot be characterized by weighted-sum method and those points turn out to have unbounded trade-offs. However, similar concepts were also proposed before and after Geoffrion. Kuhn and Tucker [START_REF] Kuhn | Nonlinear programming[END_REF] introduced a notion for proper effi-ciency in differentiable multi-objective optimization which is equivalent to Geoffrion's definition under convexity. Benson [START_REF] Benson | An improved definition of proper efficiency for vector maximization with respect to cones[END_REF] and Henig [START_REF] Henig | Proper efficiency with respect to cones[END_REF] proposed two different notions for proper efficiency in general vector optimization which are equivalent to Geoffrion's definition in multi-objective optimization. Some scalarization methods are able to find properly efficient solutions and to prove their existence. Geoffrion [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF] established that any optimal solution of the weighted sum method with a strictly positive combination of objective functions is a properly efficient solution. Benson [START_REF] Benson | Existence of efficient solutions for vector maximization problems[END_REF] introduced a new scalarization method, usually referred to as Benson's method. It is well-known under convexity that if Benson's method corresponding to a given multi-objective optimization problem is unbounded, then there exists no properly efficient solution [START_REF] Benson | Existence of efficient solutions for vector maximization problems[END_REF]. Moreover, Kaliszewski [START_REF] Kaliszewski | Quantitative Pareto Analysis by Cone Separation Technique[END_REF] introduced a scalarization method that is able to produce efficient solutions with a given bounded trade-off.

An important factor involved with multi-objective optimization problems is uncertainty. The uncertainty may arise from estimation of parameters in the model, error of machine, structure of problem and so on. Indeed, some parameters are often unknown at the beginning of solving a multi-objective optimization problem.

Three common approaches addressing this subject in the literature are stochastic optimization, robust optimization and stability/sensitivity analysis. Stochastic optimization and robust optimization incorporate uncertainty in the model. Nevertheless, in stability and sensitivity analysis methods first one solves the problem regardless of uncertainty, and then analyzes continuity and differentiability of the optimal value or the optimal set with respect to an uncertain set. This thesis is organized as follows. In Chapter 1, some preliminaries on multiobjective optimization and set-valued mappings are given. Chapter 2 is devoted to scalarization methods. The purpose of this chapter is to check the existence of properly efficient solutions by scalarization techniques. In Chapter 3, we study the concept of robustness in multi-objective optimization. We extend Georgiev et al.'s definition [START_REF] Georgiev | Robust aspects of solutions in deterministic multiple objective linear programming[END_REF] to nonlinear case and provide some theoretical results concerning this notion. Chapter 4 is dedicated to stability and sensitivity analysis in multi-objective optimization. We investigate pseudo-Lipschitz continuity and semi-differentiability of set-valued mappings corresponding to the efficient sets and efficient values of parametric multi-objective optimization problems. This is author's quick glance at the subject of the thesis. More detailed introduction is provided at the beginning of each chapter containing more information about the topic treated there. More history about multi-objective optimization can be found in [START_REF] Wallenius | Multiple Criteria Decision Making: From Early History to the 21st Century[END_REF]. In addition, [START_REF] Wiecek | Continuous multiobjective programming[END_REF] provides an overview of some works on this subject.

The results of this thesis are essentially published in the following works:

1. Soleimani-damaneh, M., and Zamani, M. On Benson's scalarization in multiobjective optimization. Optimization Letters, in press.

2. Zamani, M., Soleimani-damaneh, M., and Kabgani, A. Robustness in nonsmooth nonlinear multi-objective programming. European Journal of Operational Research 247, 2 (2015), 370-378.

3. Luc, D. T., Soleimani-damaneh, M., and Zamani, M. Stability and sensitivity analysis in multi-objective optimization, in preparation.

Originality

I certify that this work contains no material which has been accepted for any degree or diploma in my name, in any university. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person, except where due reference has been made in the text.

Chapter 1 Preliminaries

This chapter contains some background materials on set-valued mappings and multiobjective optimization. Moreover, some preliminary results are provided which will be used later. We also provide notations that will be used throughout the thesis.

In most cases, we follow the notations from the books [START_REF] Clarke | Functional Analysis[END_REF][START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Khan | Set-valued Optimization[END_REF] according to the subject.

In the present work, we consider problems in finite dimensional Euclidean spaces only. The notation k.k stands for Euclidean norm and both ha, bi and a T b denote inner product of a, b 2 R n .F o ras e tX ✓ R n ,w eu s et h en o t a t i o n sint(X), cl(X), co(X) and pos(X) for the interior, the closure, the convex hull and the convex conic hull of X,r e s p e c t i v e l y . W eu s et h en o t a t i o nx X -! x to show that x tends to x while x 2 X. Moreover, the notation t # 0 means that t goes to 0 from above. In matrix spaces, the notation k.k stands for Frobenius norm.

Multi-objective optimization

We consider the following general multi-objective optimization problem (MOP):

min f (x) s.t. x 2 X.
The set of feasible solutions of this problem is a closed set X ✓ R n and f : X ! R p is a continuous vector-valued function.

The solution notion for (MOP) is defined with respect to an ordering cone which is used for ordering the criterion space R p .W eu s et h en a t u r a lo r d e r i n gc o n ed e fi n e d by R p = = {x 2 R p : x j ≥ 0,j=1, 2,...,p}.

Utilizing this ordering cone, a feasible solution x 2 X is called an efficient solution of

(MOP) if ✓ f (x) -R p = ◆ \ f (X)={f (x)}.
Likewise, we say x 2 X is a weakly efficient solution of (MOP) if

✓ f (x) -R p > ◆ \ f (X)=;,
where R p > = {x 2 R p : x j > 0,j=1 , 2,...,p}. We will use the symbol R p ≥ to denote the set R p = \{0}. Throughout the thesis, the notations 5,  and < stand for the following orders on R p :

x 5 y () y -x 2 R p = , x  y () y -x 2 R p ≥ , x<y () y -x 2 R p > .
All above-mentioned orders are transitive. Nevertheless, only the order 5 is reflexive and antisymmetric which induces a partial order on R p .

A solution concept which plays an important role in multi-objective optimization, from both theoretical and practical points of view, is proper efficiency. This notion was introduced to eliminate the points which cannot be characterized by weightedsum method and those points turning out to have unbounded trade-offs [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF]. There are different definitions for proper efficiency in the literature; see [START_REF] Guerraggio | On the notion of proper efficiency in vector optimization[END_REF][START_REF] Sawaragi | Theory of multiobjective optimization[END_REF]. We use the following ones.

Definition 1.1. Af e a s i b l es o l u t i o nx 2 X is called properly efficient in the Geoffrion's sense, if it is efficient and there is a real number M>0 such that for all i 2{1, 2,...,p} and x 2 X satisfying f i (x) <f i (x) there exists an index j 2{1, 2,...,p} such that f j (x) >f j (x) and

f i (x) -f i (x) f j (x) -f j (x)  M. Definition 1.2. Af e a s i b l es o l u t i o nx 2 X is called properly efficient in the Benson's sense, if cl ⇣ pos(f (X)+R p = -f (x) ⌘ \ (-R p = )={0}. Definition 1.3. Af e a s i b l es o l u t i o nx 2 X is called properly efficient in the sense of Henig if (f (x) -C) T f (X)={f (x)}, for some convex pointed closed cone C satisfying R p = \{0}✓int(C).
These three definitions have been proven to be equivalent for the natural cone; see [START_REF] Sawaragi | Theory of multiobjective optimization[END_REF]. Hereafter, the set of efficient solutions, the set of weakly efficient and the set of properly efficient solutions are denoted by X E , X WE and X PE , respectively. The set Similarly, the set of weakly nondominated points, denoted by Y WN ,i sd e fi n e db y

X WE is closed but X E
Y WN = f (X WE ). The set Y is called R p = -closed, if Y + R p = is closed. Similarly, Y is called R p = -convex, if Y + R p = is convex. If X is a convex set and f 1 ,f 2 ,...,f p are con- vex functions, then Y is R p = -convex.

Tangent cones and normal cones

We start this section with a definition of set convergence. The proof of forthcoming statements in this section can be found in [START_REF] Clarke | Functional Analysis[END_REF][START_REF] Khan | Set-valued Optimization[END_REF][START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory[END_REF].

lim sup t! t C t = {x : 9{t n }✓T,9{x n }; x n 2 C tn ,t n ! t, x n ! x}, lim inf t! t C t = {x : 8{t n }✓T,t n ! t, 9{x n }; x n 2 C tn ,x n ! x}.
These limits are known as Kuratowski-Painlevé upper and lower limits, respectively. Both limits are closed sets. It is easy to see that the lower limit is a subset of the upper limit.

The main idea behind the definition of a tangent cone is to obtain a set involving all directions which one can move along with in the set. Some important applications of tangent cone in optimization theory is its vital role in deriving optimality conditions and specifying admissible directions in numerical methods. Elements of tangent cone are known as variations in calculus of variation. Tangent cone plays a role similar to tangent space in differential geometry. As in optimization, encountering nonsmoothness even for differentiable problems, the notion of tangent space is not practicable.

There are various definitions of tangent cone in the literature . Here, we just mention some of them. We refer the reader to [START_REF] Giorgi | On the notion of tangent cone in mathematical programming[END_REF] and the comprehensive book [START_REF] Khan | Set-valued Optimization[END_REF] for more detailed information about tangent cones.

Definition 1.5. The Bouligand tangent cone to X at x 2 X,d e n o t e db yT X (x),i s defined as

T X (x) := lim sup t#0 X - x t .
In other words, d 2 T X (x),i ft h e r ea r es e q u e n c e s{x n }✓X and t n # 0 such that t -1 n (x nx) ! d,o re q u i v a l e n t l y ,i ft h e r ea r es e q u e n c e sd n ! d and t n # 0 with {x + t n d n }✓X.

The Bouligand tangent cone is a nonempty closed cone, but not necessarily convex.

When X is a differentiable manifold, the Bouligand tangent cone at a given point coincides with the tangent space. Moreover, the Bouligand tangent cone is closure of admissible directions as X is convex.

Definition 1.6. Let x 2 X.T h ea d j a c e n tc o n et oX at x 2 X,d e n o t e db yT a X (x),i s defined by

T a X (x):=liminf t#0 X - x t .
In other words, d 2 T a X (x),i ff o re v e r ys e q u e n c et n # 0,t h e r ei sas e q u e n c ed n ! d with {x + t n d n }✓X.P u td i ff e r e n t l y ,d 2 T a X (x) if and only if there is a function φ :[0,✏) ! X such that φ(0) = x and φ 0

+ (0) = d.
Adjacent cone is nonempty, closed and a subset of the Bouligand tangent cone.

Definition 1.7. The Clarke tangent cone to X at x 2 X is defined by

T c X (x):= liminf t#0,x X -! x X -x t .
In other words, d 2 T c X (x) if and only if for each sequence x X -! x along with a sequence t n # 0 there is a sequence

d n ! d such that {x n + t n d n }✓X.
the Clarke tangent cone is closed and convex. Moreover, it is a subset of adjacent cone. In Section 1.4, we will give an equivalent definition for the Clarke tangent cone.

The following example demonstrates that inclusion relation among tangent cones may be strict.

Example 1.1. Let X = cl({(x 1 ,x 2 ):x 2 = sin(x -1 1 ),x 1 > 0}) [ pos( 2 4 -2 -1 3
5 ) be given.

At the origin, we have

T X (0) = {(d 1 ,d 2 ):d 1 ≥ 0}[pos( 2 4 -2 -1 3 5 ), T a X (0) = pos( 2 4 0 1 3 5 , 2 4 0 -1 3 5 ) [ pos( 2 4 -2 -1 3 5 ), T c X (0) = {0}.
Figures 1-1 and 1-2 depict X and corresponding tangent cones, respectively.

As we can see from the above example, it is generally impossible to compute the Clarke tangent cone if the Bouligand tangent or the adjacent cone is known at a

x 1 x 2 Figure 1-1 -A set related to Example 1.1 x 1 x 2 (a) T X (0) x 1 x 2 (b) T a X (0) Figure 1-2 -Tangent cones in Example 1.1
given point. For instance, tangent cone may be the whole space while the Clarke tangent cone is equal to zero. However, the following interesting theorem addresses a connection between the Bouligand tangent cone around a point and Clarke tangent cone at that point.

Theorem 1.1. Let x 2 X be a given point and let X be local ly closed around it.

Then,

T c X (x) = lim inf x X -! x T X (x).
Let K be a cone. The polar cone of K is defined and denoted by K • := {⌫ : h⌫, di0, 8d 2 K}. The polar cone of a subspace equals the orthogonal subspace.

In the rest of this section, we pay attention to the concept of normal cones. Normal cone is a counterpart of normal bundle in differential manifolds.

Definition 1.8. Let x 2 X be given and let X be local ly closed around it. The Fréchet normal cone to X at x,d e n o t e db y NX (x),i sd e fi n e da s

NX (x)={⌫ 2 R n :limsup x X -! x h⌫, x -xi kx -xk  0}.
In other words, ⌫

:l i m s u p x X -! x h⌫,x-xi kx-xk  ✏}. If X is locally closed around x,t h e nw eh a v e N X (x):=limsup x X -! x,✏#0 NX (x).
(1.1)

The preceding theorem states that the Fréchet normal cone can be replaced by N✏ (x) in the definition of limiting normal cone. Mordukhovich [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory[END_REF] introduced this set to define his normal cone by the formula (1.1). However, in finite-dimensional spaces, Mordukhovich's normal cone coincides with the limiting normal cone; see [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I: Basic Theory[END_REF] for more details.

Definition 1.9. Let X be local ly closed around x. X is said to be regular at x if

NX (x)=N X (x).
It is easily seen that X is regular at x,i fa n do n l yi fT X (x)=T c X (x).

Set-valued mappings

Let Γ:X ◆ R p be a set-valued mapping, i.e., Γ sends each x 2 X to a subset of R p .

We call the set gphΓ: ={(x, y):x 2 X, y 2 Γ(x)} as graph of Γ. Moreover, domain of Γ is defined as dom(Γ) := {x 2 X :Γ(x) 6 = ;}.

In this section, we review some notions about continuity and differentiability of setvalued mappings. The reader can find the proof of the forthcoming statements given in this section in [START_REF] Khan | Set-valued Optimization[END_REF]. Unfortunately, there are no unified terminology in this area.

Continuity

Definition 1.10. The set-valued mapping Γ is said to be closed at x 2 X,i ff o re a c h convergent sequence {(x n ,y n )}✓gphΓ with (x n ,y n ) ! (x, ȳ),w eh a v eȳ 2 Γ(x).I n other words, lim sup x!x Γ(x) ✓ Γ(x).

It is clear from the definition that if Γ is closed at x,thenthesetΓ(x) is also closed.

Definition 1.11. We say that Γ is lower semicontinuous at x 2 X,iflim inf x!x Γ(x) ◆ Γ(x),i . e . ,f o re a c ho p e ns e tU in R p with Γ(x) \ U 6 = ;,t h e r ee x i s t san e i g h b o r h o o d

O of x such that Γ(x) \ U 6 = ;, 8x 2 O.
In addition, Γ is upper semicontinuous at x 2 X,i ff o re a c ho p e ns e tU containing Γ(x),t h e r ee x i s tan e i g h b o r h o o dO of x with

Γ(x) ✓ U, 8x 2 O.
We say that Γ is continuous at x,i fi ti sb o t hu p p e rs e m i c o n t i n u o u sa n dl o w e r semicontinuous at the point. It is easily seen that upper semicontinuity of Γ at x and closedness of Γ(x) imply closedness of Γ at the point. However, the converse is not necessarily true; for instance, consider Γ:R ◆ R given by

Γ(x)= 8 > < > : {0,x -1 },x 6
(⌫, 0) 2 N gphΓ (x, ȳ) ) ⌫ =0.
The preceding theorem in known as Mordukhovich criterion in the literature. In this section, we only recalled some notions about continuity of set-valued mapping which will be used in our work. More concepts, including continuity notions defined by Pompeiu-Hausdorff distance, can be very utile and meaningful in some situations.

For a deeper discussion of these notions and their relationships with stated concepts, we refer the reader to [START_REF] Khan | Set-valued Optimization[END_REF][START_REF] Rockafellar | Lipschitzian properties of multifunctions[END_REF][START_REF] Rockafellar | Variational analysis,v o l .3 1 7[END_REF].

Differentiability

In this subsection, we review some concepts about differentiability of set-valued map-

DΓ(x, ȳ)( d)=limsup d! d,t#0 Γ(x + td) - ȳ t . (1.2)
The last formula is an extension of the upper Dini derivative notion for set-valued mappings. As the Bouligand tangent cone can be intractable in some cases (may be whole space or non-convex), other tangent cones have also been used to define new notions of derivative.

Definition 1.15. Let (x, ȳ) 2 gphΓ be given. A set-valued mapping D adj Γ(x, ȳ):

R n ◆ R p is said to be adjacent derivative of Γ at (x, ȳ),i f gphD adj Γ(x, ȳ)=T a gphΓ (x, ȳ).

In addition, we say Γ is Proto-differentiable at (x, ȳ),i fT gphΓ (x, ȳ)=T a gphΓ (x, ȳ).

Rockafellar [START_REF] Rockafellar | Proto-differentiability of set-valued mappings and its applications in optimization[END_REF] introduced the concept of Proto-differentiability to take advantage of derivability. He also established the Proto-differentiability of a class of set-valued mappings corresponding to some parametric systems (under some mild conditions).

Likewise, Clarke derivative is defined as follows.

Definition 1.16. Let (x, ȳ) 2 gphΓ be given. A set-valued mapping D c Γ(x, ȳ):R n ◆ R p is called Clarke derivative of Γ at (x, ȳ),i f

gphD c Γ(x, ȳ)=T c gphΓ (x, ȳ).
Moreover, Γ is said to be Clarke differentiable at (x, ȳ),i fT gphΓ (x, ȳ)=T c gphΓ (x, ȳ).

It is easily seen that Clarke differentiability implies Proto-differentiability. In the following example, we calculate the aforementioned derivatives at a given point.

Example 1.3. Let Γ:R ◆ R be given by

Γ(x)= 8 > < > : {t : t  xsin(lnx)},x > 0 {t : t  0},x  0. At (x, ȳ)=( 0 , 0),w eh a v eDΓ(x, ȳ)(1) = {e : e  1}, D adj Γ(x, ȳ)(1) = {e : e -1}
and D c Γ(x, ȳ)(1) = ;.

Relation (1.
2) provides a motivation for defining the following concept.

Γ(x)= 8 > < > : { p x},x > 0 {0},x  0.
At (x, ȳ)=(0, 0),w eh a v eD adj Γ(x, ȳ)(0) = {e : e ≥ 0},b u tD low Γ(x, ȳ)(0) = ;.

The following proposition provides a sufficient condition under which the converse of the above statement holds.

Proposition 1.1. Let (x, ȳ) 2 gphΓ be given and let Γ be pseudo-Lipschitz continuous at the point. If Γ is Proto-differentiable at (x, ȳ),t h e ni ti sa l s os e m i -d i ff e r e n t i a b l ea t the point.

If a set-valued mapping Γ is convex-valued and pseudo-Lipschitz continuous at a

given point, then it is semi-differentiable [START_REF] Dontchev | Implicit Functions and Solution Mappings[END_REF]. In the remainder of this section, we pay attention to the concept of strictly lower Dini derivative proposed by Thibault [START_REF] Thibault | Tangent cones and quasi-interiorly tangent cones to multifunctions[END_REF].

Definition 1.18. Aset-value dmappingD s-low Γ(x, ȳ):R n ◆ R p is said to be strictly lower Dini derivative of Γ at (x, ȳ) 2 gphΓ if for each d,

D s-low Γ(x, ȳ)( d)= liminf d! d,t#0 (x,y) gphΓ ---! (x,ȳ) Γ(x + td) -y t .
In It is easy to see that the graph of s-derivative contains that of contingent derivative. In addition, all of the derivatives presented so far will be equal when the graph of the set-valued mapping in question is convex.

We conclude the section by the celebrated Kakutani fixed-point theorem. The proof of Theorem 1.4 can be found in [START_REF] Franklin | Methods of Mathematical Economics: Linear and Nonlinear Programming[END_REF]. It is worth mentioning that John Nash used the Kakutani fixed-point theorem to prove the existence of equilibrium in finite impartial games.

Clarke subdifferential

Throughout this work, we consider locally Lipschitz data when we are talking about nonsmooth data. To handle nonsmoothness, we choose the Clarke generalized gradient. This tool was introduced by Clarke in 1973 in his Ph.D. thesis and since then has found considerable applications in both optimization and nonsmooth analysis. It shares many classical theorems of real analysis such as mean value theorem, implicit function theorem and so on. We only provide the definitions and the results that will be needed in the sequel. See [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF][START_REF] Clarke | Functional Analysis[END_REF] for more detailed information on this subject.

According to Rademacher's theorem, we know every locally Lipschitz function on R n is almost everywhere differentiable in the sense of Lebesgue measure. The definition of generalized gradient is as follows.

Definition 1.20. Let f : R n ! R p be a local ly Lipschitz function. The generalized gradient of f at x,d e n o t e db y@f(x),i sd e fi n e db y @f(x):=co{ lim

n!1 rf (x n ):x n ! x, x n / 2 X, x n / 2 X f },
where X is any set with zero Lebesgue measure and X f is the set of points at which

f is not differentiable. Definition 1.21. Let f : R n ! R p .W es a yf is strictly differentiable at x if there is rf (x) 2 R p⇥n such that lim x,y!x f (x) -f (y) -hrf (x),x-yi kx -yk =0.
It is readily seen that continuously differentiable functions are strictly differentiable on their domains. Generalized gradient enjoys nice properties. We list some of them in the following propositions. 1) @f(x) is a nonempty, convex and compact set.

2) For every ⌫ 2 @f(x) we have k⌫kk.

3) The set-valued mapping @f(.) is upper semi-continuous and closed.

4) f is strictly differentiable at x if and only if @f(x) is a singleton.

The next proposition provides more properties when f is from R n to R.

Proposition 1.4. Suppose that f : R n ! R is a locally Lipschitz function.

1) If f is convex, then generalized gradient coincides with subgradient in the sense of classic convex analysis.

2) If x is a local extremum, then 0 2 @f(x).

3)

lim sup x!x,t#0 f (x + td) -f (x) t =m a x ⌫2@f(x)
h⌫, di.

We denote the value max ⌫2@f(x) h⌫, di by f • (x; d).

Definition 1.22. Let f : R n ! R be a local ly Lipschitz function. f is called regular at x,i ff o re a c hd 2 R n , f 0 (x; d) := lim t#0 f (x + td) -f (x) t =m a x ⌫2@f(x)
h⌫, di.

We cal l f : R n ! R m regular at a given point, if its components are regular at the point.

We say that a function is regular, if it is regular at each point in its domain. The class of regular functions is vast; for example, convex functions and strictly differentiable functions are regular. Furthermore, sum and pointwise maximum of a finite number of regular functions are also regular. However, negative scaler multiplication does not necessarily preserve regularity; consider f (x)=kxk.t h en e x tp r o p o s i t i o n

gives some calculus rules.

Proposition 1.5. Let f 1 ,...,f k : R n ! R be local ly Lipschitz functions.

1) For each scaler λ, @λf 1 (x)=λ@f 1 (x).

2)

@(Σ k i=1 f i (x)) ✓ Σ k i=1 @f i (x).
3) Suppose f is given by f

(x)=m a x k i=1 f i (x).T h e n ,@f(x) ✓ co([ i2I(x) @f i (x)), where I(x)={i : f i (x)=f (x)}.
The inclusions given in properties ( 2) and (3) hold as equality when f i are regular.

Theorem 1.5. (Mean Value Theorem) Let f : R n ! R be a local ly Lipschitz function. Then, for each x, y 2 R n ,t h e r ee x i s t sz 2 (x, y) such that

f (y) -f (x) 2h@f(z),y-xi. It is easy to see that if f : R n ! R p ,t h e n@f(x) ✓ 2 6 6 6 6 6 6 4 @f 1 (x) . . @f p (x)
@f(0) = co( 2 4 1 -1 3 5 , 2 4 - 1
1 3 5 ), while 2 4 @f 1 (0) @f 2 (0) 3 5 = co( 2 4 -1 -1 3 5 , 2 4 -1 1 3 5 , 2 4 1 -1 3 5 , 2 4 1 1 3 5 ). Theorem 1.6. (Implicit Function Theorem) Let f : R n ⇥ R p ! R p be a local ly
Lipschitz function and let f (x, ȳ)=0.S u p p o s et h a te a c hM 2 @ y f (x, ȳ) is invertible.

Then, there exist a neighborhood O of x and a Lipschitz function

g : O ! R p such that g(x)=ȳ and f (x, g(x)) = 0, 8x 2 O.
In the above theorem, is invertible for each M 2 @f(x).

@ y f (x, ȳ)={M : 9N ;[NM ] 2 @f(x, ȳ)}.A
The answer to Question 1.1 for some classes of functions is affirmative; see [START_REF] Dorsch | On the local representation of piecewise smooth equations as a lipschitz manifold[END_REF] and the references therein. The response to this query may lead to extending important results about Lipschitz manifolds and implicit function theorems. Consequently, some results in sensitivity analysis could be extended. Now, let us see the implicit function theorem in a more general setting [START_REF] Jeyakumar | Nonsmooth Vector Functions and Continuous Optimization[END_REF][START_REF] Rockafellar | Lipschitzian properties of multifunctions[END_REF].

Theorem 1.7. Let f : R q ⇥R n ! R p be local ly Lipschitz. Suppose that the set-valued mapping Γ is defined by

Γ(u)={x 2 C : f (u, x) 2 K},w h e r eK ✓ R p is a closed
and convex cone and C ✓ R n is closed and convex. If 0 2 int(M (T (x)) + K) for every M 2 @ x f (ū, x),t h e nΓ is pseudo-Lipschitz continuous at (ū, x). Now, we characterize Clarke tangent cones and Clarke normal cone by generalized gradient notion. Let d X be the distance function defined by d X (y)=i n f{kyxk :

x 2 X}. The distance function is Lipschitz with modulus 1. Thanks to the distance function, Clarke tangent cone can be expressed as

T c X (x)={d : d • X (x; d)  0} and N c X (x)=cl ⇣ pos(@d X (x)
⌘ .

Lemma 1.1. The vector d belongs to Clarke tangent cone X at x 2 X if and only if for each sequence x n X -! x and t n # 0,t h e r ee x i s ts u b s e q u e n c e s{x n k } and

d n k ! d such that {x n k + t n k d n k }✓X.
Proof. Due to the facts addressed before the lemma, we only need to prove the "only if part". Let d fulfill the stated property. From the definition of generalized gradient, there are sequences y n ! x and t n # 0 such that

lim n!1 d X (y n + t n d) -d X (y n ) t n = d • X (x; d).
Of course, {y n } converges to x. Consider a sequence {x n } with ky nx n kd X (y n )+ t n /n. Therefore, there is a sequence

d n k ! d such that {x n k + t n k d n k }✓X.A sa result, d X (y n k + t n k d)  d X (x n k + t n k d n k )+ky n k -x n k k + t n k kd -d n k k  d X (y n k )+t n k kd -d n k k + t n k n k . These imply d • X (x; d)=0,l e a d i n gt od 2 T c X (x).
In most practical problems, sets are characterized explicitly by functions. The next theorem provides estimations for tangent cones and normal cones in these situations.

Theorem 1.8. Let X ✓ R n be given by X = {x : g(x)50},w h e r eg : R n ! R m is a locally Lipschitz function. Suppose that x 2 X and I(x): ={i :

g i (x)=0 }.I f 0 / 2 co([ i2I(x) @g i (x)),t h e n {d : g • i (x; d)  0, 8i 2 I(x)}✓T c X (x), N c X (x) ✓ co({@g i (x), 8i 2 I(x)}).
In addition to the above-mentioned assumptions, if g is regular at x,t h e ne q u a l i t y holds in both inclusions, X is regular at x and we have T X (x)=T c X (x).

Proof. Let h be given by h(x)=max 15i5m g i (x). Thus, X = {x : h(x) 5 0} and the theorem follows from Theorem 10.42 in [START_REF] Clarke | Functional Analysis[END_REF] and Proposition 1.5.

We close this chapter by the extended Farkas Lemma for semi-infinite linear systems.

Theorem 1.9. [START_REF] Goberna | Linear Semi-Infinite Optimization[END_REF] Let T be a given index set. Then, exactly one of the fol lowing two statements is true:

(i) The following system has a solution.

> <

> :

a T t x  0,t2 T a T x>0.
(ii) a 2 cl(pos({a t : t 2 T })).

Chapter 2

Scalarization: Extending two important theorems

Introduction

An important question in multi-objective optimization is the investigation of existence of proper efficient solutions. Scalarization methods are not only powerful tools to generate efficient solutions, but also provide valuable information about proper and improper efficient solutions. Here, we only concentrate on the applications of scalarization tools related to properly efficient solutions. We refer the reader to [START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Eichfelder | Adaptive Scalarization Methods in Multiobjective Optimization[END_REF][START_REF] Marler | Survey of multi-objective optimization methods for engineering[END_REF] for studying more aspects of these methods.

By scalarization methods, one formulates a single objective optimization problem corresponding to a given multi-objective optimization problem and he/she studies the relationships between the solutions of the two problems. It turns out in multiobjective optimization that any efficient solution of a given multi-objective optimization problem can be characterized as a solution of certain single objective optimization problems [START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Eichfelder | Adaptive Scalarization Methods in Multiobjective Optimization[END_REF].

Scalarization techniques have numerous applications in interactive algorithms [START_REF] Miettinen | Nonlinear Multiobjective Optimization[END_REF],

estimating Pareto frontier [START_REF] Marler | Survey of multi-objective optimization methods for engineering[END_REF], obtaining solutions with bounded trade-offs [START_REF] Kaliszewski | Quantitative Pareto Analysis by Cone Separation Technique[END_REF], etc.

In this chapter, we review some popular scalarization methods, though most of them are closely related. The reader is referred to the book [START_REF] Eichfelder | Adaptive Scalarization Methods in Multiobjective Optimization[END_REF] for more details. In the chapter, we consider the following general multi-objective optimization problem:

min f (x) s.t. x 2 X, (2.1)
where X is a subset of R n and f : X ! R p .

Benson's method

This section deals with the Benson's method as a popular scalarization technique.

Benson's method [START_REF] Benson | Existence of efficient solutions for vector maximization problems[END_REF] gives an examination of the existence of efficient and properly efficient solutions for multi-objective optimization problems. This technique employs l 1 -norm to check the efficiency of a given point x 0 by solving the following singleobjective optimization problem:

max p X k=1 l k s.t. f k (x 0 ) -l k -f k (x)=0,k =1, 2,...,p, l k ≥ 0,k =1, 2,...,p, x 2 X. (2.2)
A linear version of the above model was studied by Ecker and Kouada [START_REF] Ecker | Finding efficient points for linear multiple objective programs[END_REF]. Some useful examples clarifying Model (2.2) can be found in Benson [START_REF] Benson | Existence of efficient solutions for vector maximization problems[END_REF] and Giannessi et al. [START_REF] Giannessi | On the theory of vector optimization and variational inequalities. image space analysis and separation[END_REF].

The vector x 0 2 X is efficient if and only if the optimal value of Problem (2.2) is zero; see [START_REF] Benson | An improved definition of proper efficiency for vector maximization with respect to cones[END_REF][START_REF] Ehrgott | Multicriteria optimization[END_REF]. If (x, l) is an optimal solution of Model (2.2), then x is an efficient solution; see [START_REF] Benson | An improved definition of proper efficiency for vector maximization with respect to cones[END_REF][START_REF] Ehrgott | Multicriteria optimization[END_REF]. As can be seen from the above theorem and discussion, any optimal solution of Problem (2.2) yields an efficient solution. Furthermore, in many cases the unboundedness of Problem (2.2) shows that no properly efficient solution exists. In the following, we prove that this important result holds in all cases (without any assumption).

Theorem 2.2. If Problem (2.2) is unbounded, then X PE = ;. Proof. If x 0 2 X is efficient, then Problem (2.
2) has a finite optimal value equal to zero (see [START_REF] Benson | Existence of efficient solutions for vector maximization problems[END_REF][START_REF] Ehrgott | Multicriteria optimization[END_REF]). Therefore, due to the assumption, we have x 0 2 X\X E .

To the contrary, assume that there exists x 2 X PE .S i n c ex 2 X PE ,d u et ot h eH e n i g proper efficiency, there exists a convex and pointed cone C,s u c ht h a tR p ≥ ✓ intC and

(f (x) -C \{0}) \ f (X)=;. According to R p ≥ ✓ intC,w eh a v ee i 2 intC,f o ri =1 , 2,.
..,m (vector e i denotes the ith unit vector in R p ). Thus, there exists r i > 0 such that B(e i ; r i )={y 2 R p : kye i k <r i }✓C. Now, consider the following system, in which ↵, ✓,a n dy are variables:

8 > > > < > > > : f (x 0 ) -↵e i = f (x) -✓(e i + y), kyk <r i , ↵>0,✓>0. (2.3) Set ↵i := 2r -1 i kf (x) -f (x 0 )k and ȳ := r i 2 ⇥ f (x)-f (x 0 ) kf (x)-f (x 0 )k . The vector (↵, ✓, y)=(↵ i , ↵i , ȳ)
is a solution of the system (2.3), and e i + y 2 B(e i ; r i ). Thus,

f (x 0 ) -↵i e i -f (x) 2-C.
Also, clearly we have

8↵> ↵i , -(↵ -↵i )e i 2-C \{0}.
Therefore, for each ↵> ↵i ,w eh a v ef

(x 0 ) -↵e i -f (x) 2-C \{0},b e c a u s eC is a convex cone. This implies f (x 0 ) -↵e i 2 f (x) -C \{0}.H e n c e ,f (x 0 ) -↵e i / 2 f (X),
due to Henig proper efficiency.

So far, we proved that for each e i ,i=1, 2,...,m,thereexistsapositivescalar↵ i > 0, such that

8↵>↵ i ,f (x 0 ) -↵e i / 2 f (X). (2.4)
Now, defining ↵ := max 1im ↵i ,w esho wthatforan yd 2 R p = satisfying P m j=1 d j =1 and any ↵>↵,w eh a v ef (x 0 ) -↵d / 2 f (X). The cone -C is convex and also, for

each i,w eh a v ef (x 0 ) -↵e i -f (x) 2-C. Therefore, m X i=1 d i (f (x 0 ) -↵e i -f (x)) 2-C.
This implies

f (x 0 ) -↵d -f (x) 2-C.
Furthermore, we have d 2 C, which implies

-(↵ -↵)d 2-C \{0}, 8↵>↵.
Thus,

f (x 0 ) -↵d -f (x) 2-C\{0}, 8↵>↵.
Therefore, due to the proper efficiency of x,w eg e t

f (x 0 ) -↵d / 2 f (X), 8↵>↵.
Thus, ↵ provides an upper bound for the objective function of Problem (2.2). This contradicts the unboundedness assumption on Problem (2.2), and completes the proof.

It is worth mentioning that the converse of Theorem 2.2 does not hold necessarily.

The following example illustrate this point.

Example 2.1. Consider this multi-objective optimization problem.

min (x 1 ,x 2 ) s.t. x 1 x 2 =1, x 2-R 2 > .
It is easy to see that the above problem does not have any properly efficient solution.

However, the Benson's problem for each x 0 belonging to feasible set is bounded. Now, we consider the following single-objective problem which has been studied by Guddat et al. [START_REF] Guddat | Multiobjective and Stochastic Optimization Based on Parametric Optimization,v o l .2 6 . A k a d e m i e -V e r l a g[END_REF] as a hybrid scalarization method; see also [START_REF] Ehrgott | Multicriteria optimization[END_REF]:

min p X k=1 λ k f k (x) s.t. f k (x)  f k (x 0 ),k =1, 2,...,p, x 2 X. (2.5)
In this problem, λ 1 ,λ Under the R m = -closedness and R m = -convexity assumptions, Benson [START_REF] Benson | Existence of efficient solutions for vector maximization problems[END_REF] proved that, the unboundedness of Problem (2.2) implies X E = ;. Due to the above discussion, it can be seen that, under R m = -closedness and R m = -convexity assumptions, the unboundedness of Problem (2.5) implies X E = ; as well. Although, in Theorem 2.2, we omitted the R m = -convexity assumption for investigating X PE , the following example shows that one cannot omit this assumption for

X E . Example 2.2. Let X = {(x 1 ,x 2 ):x 1 < 0,x 2  0,x 2 ≥ 1 x 1 }[{(0, 0)} and f 1 (x)= x 1 ,f 2 (x)=x 2 .H e n c e ,Y = X is R 2 = -closed, while it is not R 2 = -convex. Considering x 0 =(0, 0),P r o b l e m( 2 . 2 )i su n b o u n d e dw h i l eX E = {(x 1 ,x 2 ) 2 X : x 2 = 1 x 1 }.
It is worth mentioning that Theorem 2.2 holds for any ordering cone. Let C ✓ R p be a convex, closed and pointed cone with int(C) 6 = ;.W e c a l lx 2 X to be an

efficient solution of Problem (2.1) with respect to C,i ff (X) \ (f (x) -C)={f (x)}.
Likewise, x 2 X is called a weakly efficient solution of Problem (2.1) with respect to

C if f (X) \ (f (x) -int(C)) = ;.
The point x 2 X is called a properly efficient solution of (2.1) in the Benson's sense with respect to C,i f

cl(pos(f (X)+C -f (x))) \ (-C)={0}.
That is equivalent to the existence of a pointed, closed and convex cone K with

C \{0}✓int(K) such that f (X) \ (f (x) -K)={f (x)} (proper efficiency in the Henig's sense).
Consider the following scalarization corresponding to Problem (2.1) with respect to C: and

min λ T f (x) s.t. f (x 0 ) -f (x) 2 C (2.6) x 2 X, in which λ 2 C • is
✓ f (X) -f (x) ◆ \ (-K)={0}. (2.7)
As Problem (2.6) is unbounded, there is a sequence {x n }✓X such that

0 6 = f (x 0 ) -f (x n ) 2 C ✓ K, λ T f (x n ) -n, 8n 2 N.
The sequence f (xn)-f (x 0 ) kf (xn)-f (x 0 )k is bounded in R n , and so, without loss of generality we may assume that this sequence converges to a nonzero d 2-C due to the closedness

of C. Moreover, it is trivial that kf (x n ) -f (x 0 )k!1 ,a sλ is fixed in R n .S o , f (xn)-f (x) kf (xn)-f (x 0 )k ! d.F u r t h e r m o r e , C \{0}✓int(K), which leads to -d 2 int(K).
Therefore, for n 0 sufficiently large, we have

f (x n ) -f (x) kf (x n ) -f (x 0 )k 2-K, 8n ≥ n 0 .
This contradicts (2.7), and completes the proof.

Direction-based scalarization

Another popular scalarization method is direction-based scalarization [START_REF] Eichfelder | Adaptive Scalarization Methods in Multiobjective Optimization[END_REF]. The directionbased scalarization problem is as follows:

min t s.t. f (x) 5 a + tr (2.8) x 2 X,
where a 2 R p and r 2 R p ≥ . This is also known as the Pascoletti-Serafini scalarization in the literature. In addition, ✏-constraint scalarization is a special case of this method [START_REF] Eichfelder | Adaptive Scalarization Methods in Multiobjective Optimization[END_REF]. Proof. As Problem (2.8) is unbounded, there is some x 2 X such that the following problem is unbounded:

Proposition 2.1. (i) If (x,
min t s.t. f (x) 5 f (x)+tr x 2 X.
Thus, the following problem is unbounded:

max p X i=1 l i s.t. f (x)+l = f (x) x 2 X, l = 0.
By virtue of Theorem 2.2, the properly efficient set of multi-objective optimization problem (2.1) is empty.

Compromise Programming

Compromise programming is related to some useful techniques in multi-objective programming which are seeking for solutions as close as possible to an utopia point [START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Gearhart | Compromise solutions and estimation of the noninferior set[END_REF]. We call the vector y U 2 R p an utopia point, if ). A popular measure function, which has been widely used in the literature, is defined by d(λ, y)=kλ yk q , for each (λ, y) 2 R p ⇥ R p , in which q is a positive integer, λ y =(λ 1 y 1 ,λ 2 y 2 ,...,λ q y q , ) and

y U i < min x2X f i (x),f o re a c h i. The general form of a compromise programming problem corresponding to (MOP) is {min d(f (x),y U ) s.t. x 2 X}, in which d is a measure function from R p ⇥ R p to [0, +1
kλ yk q = ✓ p X j=1 |λ j y j | q ◆ 1 q .
Considering a λ 2 R p > ,t h es e to fbe s ta p p r o x i m a t i o n so ft h ei d e a lpo i n tm e a s u r e db y k.k q is defined by

A(λ, q, Y )= ⇢ ȳ 2 Y : kλ (ȳ -y U )k q =min y2Y kλ (y -y U )k q ,
in which Y = f (X). Now, the set of best approximations of y U considering all positive weights is defined by

A(Y )= [ λ2ri(∆) [ 1q<1 A(λ, q, Y ),
where ∆ stands for the standard simplex. The following result has been proved by Gearhart [START_REF] Gearhart | Compromise solutions and estimation of the noninferior set[END_REF], and it has been addressed by Ehrgott [START_REF] Ehrgott | Multicriteria optimization[END_REF] and Sawaragi et al. [START_REF] Sawaragi | Theory of multiobjective optimization[END_REF].

Theorem 2.4. If Y is R m = -closed, then A(Y ) ✓ Y PN .
Gearhart [START_REF] Gearhart | Compromise solutions and estimation of the noninferior set[END_REF] used Theorem 2.4 to show that the set of compromise solutions is dense in the set of efficient points. Another proof for this theorem has been given

by Ehrgott [START_REF] Ehrgott | Multicriteria optimization[END_REF] and Sawaragi et al. [START_REF] Sawaragi | Theory of multiobjective optimization[END_REF]. The proof given in [START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Sawaragi | Theory of multiobjective optimization[END_REF] is completely different to that provided in [START_REF] Gearhart | Compromise solutions and estimation of the noninferior set[END_REF], though the proof given in [START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Sawaragi | Theory of multiobjective optimization[END_REF] is not correct. In ap a r to ft h ep r oo f ,t h ea u t h o r sc l a i m e dt h a t

kλ (ŷ - d β k -y U )k q < kλ (ŷ -y U )k q , 8k implies lim k-! +1 kλ (ŷ - d β k -y U )k q < kλ (ŷ -y U )k q .
(See p. 188 of Ehrgott [START_REF] Ehrgott | Multicriteria optimization[END_REF].) Clearly, this assertion is not correct. In fact, < is converted to  after operating limit. This makes the given proof in [START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Sawaragi | Theory of multiobjective optimization[END_REF] incorrect.

In the following, we provide another proof for Theorem A.F u r t h e r m o r e , i t i s shown that the R m = -closedness assumption is redundant and it is removed. This assumption has been considered in the three publications [START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Gearhart | Compromise solutions and estimation of the noninferior set[END_REF][START_REF] Sawaragi | Theory of multiobjective optimization[END_REF]. In fact, Theorem A establishes that, under R m = -closedness assumption, each compromise solution is properly efficient. In the following, we show that this holds in the general case.

Theorem 2.5. A(Y ) ✓ Y PN .
Proof. Without loss of generality, we assume that y U =0. Thus, for every y 2 Y ,w e have y>0. Considering ȳ 2 A(Y ),t h e r ea r eλ 2 Λ 0 and q 2 [1, 1) such that ȳ is an optimal solution (a minimizer) of the following optimization problem:

min m X i=1 λ q i y q i (2.9a) s.t. y 2 Y. (2.9b)
We define a real-valued function f : R p = -! R,b yf (y)= P p i=1 λ q i y q i . This function has the following properties on R p > (and immediately on Y ):

(i) rf (y) > 0, 8y 2 R p > , (ii) f (y + y 0 ) ≥ f (y)+f (y 0 ), 8y, y 0 2 R p > .
The proof of property (i) is straightforward. So, we prove only property (ii). For two positive scalars y i and y 0 i ,w eh a v e(y i + y 0 i ) q ≥ (y i ) q +(y 0 i ) q . Therefore,

(λ i ) q (y i + y 0 i ) q ≥ (λ i ) q (y i ) q +(λ i ) q (y 0 i ) q 8 i =1, 2,...,m ) m X i=1 (λ i ) q (y i + y 0 i ) q ≥ m X i=1 (λ i ) q (y i ) q + m X i=1 (λ i ) q (y 0 i ) q ) f (y + y 0 ) ≥ f (y)+f (y 0 ).
β k (y k + d k -ȳ)=-d, (2.10)
for some nonzero d 2 R p = .

Note that {β k } is a sequence of positive real numbers. Therefore, this sequence is either bounded or unbounded. We consider these two possible cases and show that in both cases we have a contradiction.

Case 1. {β k } is bounded.

In this case, {β k } has a convergent subsequence. For simplicity, we denote this subsequence by {β k } again. So,

β k ! β,f o raβ ≥ 0.
Assume that β =0 . There exists an i such that d i > 0. Thus, by (2.10), there

exists k 0 2 N such that 8k ≥ k 0 ,β k (y ik + d ik -ȳi ) < - d i 2 .
Hence, for each k ≥ k 0 ,w eg e t

y ik < - d i 2β k +ȳ i .
This implies y ik !-1. This yields a contradiction, because y>0 for each y 2 Y . Now, assume that β>0.A c c o r d i n gt o( 2 . 1 0 ) ,t h e r ee x i s t san o n z e r od 0 2 R p = such that

lim k!1 y k + d k -ȳ = -d 0 . (2.11) Since f ( d 0 2 ) > 0,t h e r ee x i s t sa n✏>0 such that 8✏ 2 (0, ✏],f ( d 0 2 ) > ✓ (1 + ✏) q -1 ◆ f (ȳ).
(2.12)

Due to ȳ>0 and (2.11), there exists k 1 2 N such that for each k ≥ k 1 ,w eh a v e

y k + d k -ȳ<- d 0 2 +✏ȳ.
Hence, we get

0 <y k + d 0 2 < (1 + ✏)ȳ, which implies f (y k )+f ( d 0 2 )  f (y k + d 0 2 ) < (1 + ✏) q f (ȳ), (2.13)
because of property (ii) and the fact that f is strictly increasing. Therefore, for each

k ≥ k 1 ,w eg e t f (y k ) < (1 + ✏) q f (ȳ) -f ( d 0 2 ) <f(ȳ),
because of (2.12). This contradicts the optimality of ȳ for the optimization problem

(3.21). Case 2. {β k } is unbounded.
Without loss of generality, we assume that β k !1. Thus, from (2.10), we have

lim k!1 y k + d k -ȳ =0.
Since ȳ is an optimal solution of Problem (3.21) and d k = 0,b ypropert y(ii),w eha v e f (y k + d k ) ≥ f (ȳ).F u r t h e r m o r e ,f is a continuously differentiable function on R p > . Therefore, by the mean value theorem, we have

0  f (y k + d k ) -f (ȳ)=rf (z k ) T (y k + d k -ȳ), in which z k = ↵ȳ +(1-↵)(y k + d k ) for an ↵ 2 (0, 1). Thus, rf (z k ) T ✓ β k (y k + d k -ȳ) ◆ ≥ 0. (2.14) Since y k + d k ! ȳ,w eh a v ez k ! ȳ. Therefore, 0  lim k!1 rf (z k ) T ✓ β k (y k + d k -ȳ) ◆ = rf (ȳ) T (-d).
This is a contradiction, because rf (ȳ) > 0 and

d 2 R p = \{0} gives rf (ȳ) T (-d) < 0.
Hence, in both possible cases we had a contradiction, and the proof is complete.

Corrolary 2.2. If f (x 0 ) 2 A(Y ),t h e nt h eo p t i m a lv a l u eo fP r o b l e m( 2 . 2 )i sz e r o .

Also, x 0 is an optimal solution of Problem (2.5).

The converse of the above corollary is not correct necessarily, i.e., there exists some x 0 2 X for which the optimal value of Problem (2.2) is zero, while f (x 0 ) / 2 A(Y ).

To show this, it is sufficient to consider an improp erly efficient p oint. But if Y is R m = -closed, there exists the utopia point and the optimal value of Problem (2.2) is zero, then f (x 0 ) is the limit of some outcomes belonging to A(Y ). This results from the fact that, under the considered assumptions, the set of compromise solutions is dense in the set of efficient points; see [START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Gearhart | Compromise solutions and estimation of the noninferior set[END_REF].

Chapter 3 Robustness

Introduction

Due to perturbations and partial knowledge, in most practical optimization problems we are faced with uncertainty. Popular approaches for dealing with uncertainty are stochastic optimization, robust optimization, stability and sensitivity analysis. Each approach has its own advantages. Stability and sensitivity analysis are used to analyze as o l u t i o ns e t . I nt h efi r s ts t e p ,w en e g l e c tu n c e r t a i n t yi nt h em o d e l ,a n ds t a b i l i t y and sensitivity analysis tools are utilized after obtaining an optimal solution. They are also called post-optimal analysis techniques in the literature. We investigate robustness in the present chapter and stability and sensitivity analysis in the next chapter.

Unlike stability and sensitivity analysis, stochastic optimization and robust optimization entangle uncertainty in the first step. These approaches find solution(s) while uncertainty is involved in objective function and constraints. Consider the following uncertain multi-objective optimization problem:

min f (u, x) s.t. g(u, x) 5 0 (3.1)
x 2 X, u 2 U, where x and u are decision variable and uncertain parameter, respectively. In addition, f : U ⇥ X -! R p , g : U ⇥ X -! R m and U ✓ R q . Moreover, X ✓ R n and p ≥ 2. The set U is called an uncertain set and its members are called as scenarios. To handle the problem, we are faced with two challenges. First, which decision variable vector is admissible? Second, which admissible decision variable vector is efficient?

One practical method to address these questins is to apply scalarization techniques in order to reduce the uncertain multi-objective problem to an uncertain single objective problem. Then, one can apply the rich theory of stochastic optimization and robust optimization. The reader is referred to great books [START_REF] Ben-Tal | Robust Optimization[END_REF][START_REF] Shapiro | Lectures on stochastic programming: modeling and theory[END_REF] for details about stochastic optimization and robust optimization in single objective optimization. However, we merely mention methods that sustain the multi-objective nature of the problem.

Stochastic multi-objective optimization

In stochastic optimization, data of Problem (3.1) are assumed to have a statistical structure. We mean that there are probability measure P and σ-algebra F ✓ 2 U such that (U, F, P ) is a probability space. Moreover, for every x 2 X,thefunctionsf i (., x)

and g i (., x) are measurable (random variables).

We review two intuitive ways for managing the problem called multi-ob jective method and stochastic method. Suppose that the feasible set is a fixed set X;f o r instance, X = {x 2 X : P (g i (u, x)  0) ≥ 1 -✏, 8i} for some tolerance ✏ 2 (0, 1).

In multi-objective approach corresponding to each objective function f i ,av e c t o r

(z (1)
i (x),...,z

(is) i (x)) := (H 1 (f i (u, x)),...,H s (f i (u, x))),
is defined. Each H i can be expectation value, risk measure, etc. According to the transformation, x is called efficient if it is an efficient solution of the following multiobjective problem:

min (z (1)
1 (x),...,z

1 (x),...,z 1 p (x),...,z (ps) p (x))

s.t. x 2 X.
Stochastic methods use some aggregation function A : R p ! R and transforms the stochastic multi-objective optimization to the following stochastic single objective optimization:

min A(f 1 (u, x),...,f p (u, x)) s.t. x 2 X,
and manage this problem. More details and various stochastic methods can be seen in the two survey papers [START_REF] Abdelaziz | Solution approaches for the multiobjective stochastic programming[END_REF][START_REF] Gutjahr | Stochastic multi-objective optimization: a survey on non-scalarizing methods[END_REF].

Robustness: an introduction

Robust optimization has been introduced by Soyster [START_REF] Soyster | Convex programming with set-inclusive constraints and applications to inexact linear programming[END_REF] in 1973; However, it has attracted attention in the late 1990's. The works of Ben-tal, El Ghaoui and coworkers introduced robust optimization as a strong tool to handle uncertainty. Unlike stochastic optimization, the uncertain set U can be any arbitrary set. However, it is common to regard uncertain set as a compact set. The reader is referred to [START_REF] Ben-Tal | Robust Optimization[END_REF][START_REF] Bertsimas | Theory and applications of robust optimization[END_REF] for more information.

Consider problem (3.1) with p =1 . A common approach to deal with it is to manage the optimization problem,

min (sup u2U f (u, x)) s.t. g(u, x) 5 0, 8u 2 U, (3.2)
called robust counterpart. As the problem signifies, we try to find a point that hedges against all scenarios and simultaneously minimizes the worst case of the objective function with respect to U .

The robust counterpart problem, in general, may be intractable. For example, consider the case that U is not finite, which is a semi-infinite programming problem.

However, for some practical cases robust counterpart is manageable. For instance, in the linear case, when uncertain set is an ellipsoid or polyhedral the robust counter-part will be conic programming or linear programming, respectively. Thus, robust counterpart enjoys not only nice properties, but also there are polynomial-time algorithms to solve it; see [START_REF] Ben-Tal | Robust Optimization[END_REF][START_REF] Bertsimas | Theory and applications of robust optimization[END_REF] for more details. Semi-infinite programming tools have also been used to handle robust counterpart; see [START_REF] Goberna | Robust solutions of multiobjective linear semi-infinite programs under constraint data uncertainty[END_REF] and the references therein.

In the approach by robust counterpart we consider only feasible points that satisfy all constraints for all scenarios. In certain cases these feasible points hardly exist, and therefore the methods that disregard rare scenarios are often proposed to solve the problem (light robustness and some other notions go in this direction). We refer the reader to [START_REF] Goerigk | Algorithm engineering in robust optimization[END_REF] and reference therein for more discussions on other robust approaches.

Robustness in single objective optimization has received considerable attention in the past decades, but robustness in multi-objective optimization has been considered seriously in recent years; see [START_REF] Deb | Introducing robustness in multi-objective optimization[END_REF][START_REF] Ehrgott | Minmax robustness for multiobjective optimization problems[END_REF][START_REF] Goberna | Robust solutions to multi-objective linear programs with uncertain data[END_REF][START_REF] Goberna | Robust solutions of multiobjective linear semi-infinite programs under constraint data uncertainty[END_REF]. This notion has been studied from different standpoints. In the rest of this section, we review some concepts of robustness in multi-objective optimization.

Deb and Gupta [START_REF] Deb | Introducing robustness in multi-objective optimization[END_REF] introduced two definitions for robustness. In the first definition, they call an efficient solution robust if it is an efficient solution of the mean of all objective functions. In the second definition, the objectives do not change, but a constraint is added which restricts the absolute difference between the mean and the original objective values. There, the efficient solutions of the modified problem are called robust efficient.

Consider the following uncertain multi-objective optimization problem,

min f (u, x) s.t. x 2 X, u 2 U.
where f : U ⇥X ! R p is continuous. We only care about uncertainty in the objective function and the feasible set is not subject to uncertainty. Otherwise, we can specify asetX by the single objective robust optimization methods, such as hedging against all scenarios and so on. If U is singleton, then the problem reduces to a deterministic multi-objective problem. We denote the above problem by P (u) for a given u 2 U .

We follow the terms and the notations used in [START_REF] Ide | Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts[END_REF].

Definition 3.1. [START_REF] Ide | Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts[END_REF] The feasible solution x 2 X is called a flimsily robust (weakly) efficient solution if it is a (weakly) efficient solution of P (u) for some u 2 U . Definition 3.2. [START_REF] Ide | Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts[END_REF] We say x 2 X is highly robust (weakly) efficient if it is a (weakly) efficient solution of P (u) for every u 2 U .

It is easy to see that highly robust (weakly) efficiency implies flimsily robust (weakly) efficiency. It is likely that highly robust efficient solution may not exist even for finite uncertain set U . The following example demonstrates this.

Example 3.1. Let X = {x 1 ,x 2 ,x 3 } and U = {u 1 ,u 2 }.S u p p o s et h a tt h ev a l u e so f f : U ⇥ X ! R 2 are given as follows:

f (u 1 ,x 1 ) f (u 1 ,x 2 ) f (u 1 ,x 3 ) f (u 2 ,x 1 ) f (u 2 ,x 2 ) f (u 2 ,x 3 ) (1, 1) (3.5, 2) (3.2, 2.8) (6, 1.5) (2.6, 0.7) (5.1, 1.2)
It is readily seen that x 1 and x 2 are flimsily robust efficient while there is no highly robust efficient solution (see Figure 3-1).

f (u 1 ,x 1 ) f (u 1 ,x 2 ) f (u 1 ,x 3 ) f (u 2 ,x 1 ) f (u 2 ,x 2 ) f (u 2 ,x 3 ) Figure 3-1 -Image of f in Example 3.1
Goberna et al. [START_REF] Goberna | Robust solutions to multi-objective linear programs with uncertain data[END_REF] consider uncertain linear multi-objective optimization, and provide some necessary and sufficient conditions for highly robust weak efficiency.

Ehrgott et al. [START_REF] Ehrgott | Minmax robustness for multiobjective optimization problems[END_REF] extended the worst case robustness notion from the single objective optimization to the multi-objective case based on the idea of set order.

Definition 3.3. The decision variable x 2 X is called a set-based minmax robust efficient solution if there is no x 2 X such that

f U (x) ✓ f U (x) -R p ≥ ,
where f U (x)={f (u, x):u 2 U }.L i k e w i s e ,x 2 X is called a set-based minmax robust weakly efficient solution if there is no x 2 X such that

f U (x) ✓ f U (x) -R p > .
For the case p =1and under the compactness of U ,t h e s et w oc o n c e p t sa r e equivalent to the nonexistence of x 2 X with sup u2U f (u, x) < sup u2U f (u, x), which is exactly the definition of robustness in single objective optimization. In Example 3.1, all admissible points are set-based minmax robust efficient.

The following propositions, by taking advantage of two scalarization techniques, introduce some methods to obtain a set-based minmax robust (weakly) efficient solution. Their proofs can be found in [START_REF] Ehrgott | Minmax robustness for multiobjective optimization problems[END_REF].

Proposition 3.1. Let x be an optimal solution of the fol lowing problem:

min(sup u2U hλ, f (u, x)i) s.t. x 2 X,
where λ 2 R p > (λ 2 R p ≥ ). Then x is a set-based minmax robust (weakly) efficient solution.

Proposition 3.2. suppose that x is an optimal solution of the following problem:

min(sup u2U f i (u, x)) s.t. f j (u, x)  ✏ j j 6 = i 8u 2 U,
x 2 X, Definition 3.5. [START_REF] Fliege | Steepest descent methods for multicriteria optimization[END_REF][START_REF] Kuroiwa | On robust multiobjective optimization[END_REF] If x is an efficient solution of the following problem: Other concepts have also been proposed for robustness; we listed here some of them.

min f max U (x) x 2
For example, on account of the definition of set-based minmax robust efficiency Ide et al. [START_REF] Ide | Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations[END_REF][START_REF] Khan | Set-valued Optimization[END_REF] introduced order-robust definition by using orders for set. Schöbel et al. [START_REF] Ide | Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts[END_REF] extended light robustness to multi-objective optimization. Pourkarimi et al.

introduced two new concepts for robustness in linear multi-objective optimization [START_REF] Pourkarimi | Robustness in deterministic multi-objective linear programming with respect to the relative interior and angle deviation[END_REF].

We refer reader to the survey pap er [START_REF] Ide | Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts[END_REF].

In a recent work, Georgiev et al. [START_REF] Georgiev | Robust aspects of solutions in deterministic multiple objective linear programming[END_REF] have defined the robustness for linear multiobjective optimization problems from a different point of view. They considered a perturbation standpoint, and defined an efficient solution as a robust solution if it remained efficient for small perturbations of the coefficients of the objective functions.

They also studied their definition considering different kinds of perturbations, including changing the objectives' coefficients and adding a new objective function. They obtained necessary and sufficient conditions and presented various nice properties of the robust solutions in the linear case. Goberna et al. [START_REF] Goberna | Robust solutions to multi-objective linear programs with uncertain data[END_REF][START_REF] Goberna | Robust solutions of multiobjective linear semi-infinite programs under constraint data uncertainty[END_REF] extended Georgiev et al.'s definition for linear multi-objective optimization problems under perturbations of the coefficients of both the objective functions and constraints.

In the rest of our work here, we extend the definition given by Georgiev et al. to nonlinear multi-objective optimization. We show that, under the compactness of the feasible set or convexity, the set of robust efficient solutions is a subset of the set of properly efficient solutions. Some necessary and sufficient conditions for robust solutions with respect to the Bouligand tangent cone and non-ascent directions, under appropriate assumptions, are given. A robustness radius is calculated. The relationships between the robustness notion considered in this sense and some mentioned definitions are highlighted. Two kinds of modifications in the objective functions are dealt with and the relationships between the weak/proper/robust efficient solutions of the problems, before and after the perturbation, are established. Some examples, to clarify the theoretical results, are given.

In Section 3.2, robustness is defined, its relationship with proper efficiency is established, and some necessary and sufficient conditions are proved. Section 3.3 is devoted to the robustness radius calculation. Section 3.4 contains some results on connections between the newly and previously defined robustness definitions. In Section 3.5, we study some alterations of the objective functions that preserve weak/proper/robust efficiency.

Robustness

We start this section by intro ducing the concept of robust solution in nonlinear multiobjective optimization. This definition extends Definition 3.1 in Georgiev et al. [START_REF] Georgiev | Robust aspects of solutions in deterministic multiple objective linear programming[END_REF].

and f k (x) -f k (x i ) f j (x i ) -f j (x)
>M i , for each j 2{1,...,p} with f j (x i ) >f j (x).

(3.5)

Since X is compact, without loss of generality, we may assume that {x i } converges to some x 2 X.A l s o ,w ed e fi n eQ i = {j : f j (x i ) >f j (x)}. This set is nonempty because

x is efficient. Without loss of generality, by choosing an appropriate subsequence, Q i is a constant set for all i indices. So, we denote it by Q.T w oc a s e sm a yo c c u r for x;e i t h e ri ti se q u a lt ox or not. We consider these two possible cases and get a contradiction in each case.

Due to the robustness of x,t h e r ee x i s t ss o m e✏>0 such that x is an efficient solution of Problem (3.4) for any matrix C p⇥n with kCk <✏ .L e tx 6 =x.W ec a n choose the matrix with property k Cp⇥n k <✏such that

Cj (x -x) < -2δ, 8j 2 Q, (3.6)
Cj =0, 8j 2{1,...,p}\Q,

for some δ>0 ( Cj denotes the jth row of C). Since f is bounded on X,f r o m( 3 . 5 ) ,

we have f j (x i ) -! f j (x) for each j 2 Q as i -! +1. Therefore, for sufficiently large i values, we have f j (x i )f j (x) -δ<0. Hence, by (3.6), for sufficiently large i values, we get

f j (x i )+ Cj x i <f j (x)+ Cj x -δ<f j (x)+ Cj x, 8j 2 Q. (3.8)
Also, by (3.7) and due to the definition of Q,f o rs u ffi c i e n c yl a r g ei values, we have for some δ>0.A s s u m et h a tL j is the Lipschitz constant of f j on a neighborhood of

f j (x i )+ Cj x i  f j (x)+ Cj x, 8j
x.B y( 3 . 5 ) ,f o rs u ffi c i e n t l yl a r g ei values, we get

f j (x i ) -f j (x) < L k kx -x i k M i <δkx -x i k. (3.12)
Therefore, by (3.10)-(3.12) we get inequalities (3.8) and (3.9) in this case as well.

These contradict the robustness of x and the proof is complete.

The converse of the above theorem does not hold necessarily, even for the linear case; see Example 3.2 of [START_REF] Georgiev | Robust aspects of solutions in deterministic multiple objective linear programming[END_REF].

The following example shows that the compactness assumption of X in Theorem Proof. We argue by giving a contradiction. Supp ose that 0 6 = d 2 G(x) \ T X (x).

By robustness of x,thereexistsan✏>0 such that x is an efficient solution of problem for some δ>0 (e is a column vector with all components being equal to one). Since

d 2 T X (x), 9({x i }✓X, t i # 0); x i - x t i ! d. (3.14)
Therefore, from (3.13) and (3.14), for sufficiently large i,w eh a v e

C( x i - x t i ) < -δe, (3.15)
which implies Cx i + t i δe < C x. Using the mean value theorem (Theorem 1.5), for each i,w eh a v e

f (x i )=f (x)+⇠ T i (x i -x), (3.16)
where ⇠ i is an n⇥p matrix whose jth column belongs to @f j (x j i ),forsomex j i 2 (x, x i ). Thus,

f (x i )+ Cx i + t i δe < f(x)+⇠ T i (x i -x)+ C x ) f (x i )+ Cx i + t i (δe -⇠ T i ( x i -x t i )) <f(x)+ C x.
Since xj i -! x,a si -! +1,a n df is locally Lipschitz at x, by Proposition 1.3 the sequence {⇠ i } is bounded. Hence, ⇠ i -! ⇠,forsome⇠ 2 @f(x), because of Proposition 1.3. Thus, ⇠ T d  0. Therefore, for sufficiently large i values, ⇠ T i ( x i -x t i ) <δe.Thus, we get

f (x i )+ Cx i <f(x)+ C x,
which contradicts the robustness of x,a n dc o m p l e t e st h ep r oo f . The condition given in the above theorem is necessary for robustness and it is not sufficient, in general. The following example clarifies this.

Example 3.3. Consider the multi-objective optimization problem,

min (f 1 (x),f 2 (x)) s.t. x 2 R, where f 1 (x):=x, f 2 (x):= 8 > < > : -x |x| < 1, -x ( 1 3 )
|x|≥1.

Let x =2

.A tt h i sp o i n tw eh a v eT X (x)=R and G(x)={0}.I ti sn o td i ffi c u l tt o see that x =2is an efficient solution of the above problem, while for any ✏>0 it is not an efficient solution of

min (f 1 (x),f 2 (x)+ ✏ 2 x) s.t. x 2 R, because for each ✏>0,b ys e t t i n gx ✏ =m i n {-125, -1 ✏ 3 },w eh a v ef 1 (x ✏ ) <f 1 (2) and f 2 (x ✏ )  f 2 (2).
As shown by the above example, the necessary condition given in Theorem 3.2 may not be sufficient for robustness, in general. Theorem 3.3 establishes that this condition is sufficient under convexity assumption. Theorem 3.3. Let X be a closed and convex set and f i (i =1 ,...,p) be convex.

Assume that x is an efficient solution of Problem (3.3). x is a robust efficient solution of Problem (3.3) if and only if T X (x) \ G(x)={0}.

Proof. The "only if" part is derived from Theorem 3.2. For "if" part, suppose that x is not a robust efficient solution. Thus, there exist a sequence {C i } of p ⇥ n matrices and a sequence {x i }✓X such that

C i ! 0,a n d f (x i )+C i x i  f (x)+C i x.
(3.17)

Set In the first case, without loss of generality, we assume that

d i := x i - x kx i -xk . ( 3
x i ! x.F r o mt h ec o n v e x i t y of f ,f o ra n y⇠ 2 @f(x),w eh a v e , f (x i ) ≥ f (x)+⇠ T (x i -x), (3.19)
with ⇠ being an n ⇥ p matrix whose jth column belongs to @f j (x). Therefore, due to (3.17), we have

kx i -xk -1 ⇠ T (x i -x)+C i (x i -x)  0. (3.20)
Without loss of generality, we can assume that

d i ! d,forsomed 2 R p with kdk =1
and it is obvious that d 2 T X (x). Moreover, from (3.20) we conclude that d 2 G(x).

Thus d 2 G(x) \ T X (x). This gives a contradiction. Now, we consider the other case: {x i } does not have any subsequence convergent to

x. Therefore, without loss of generality, there exists a r>0 such that kx ixk >r.

On the other hand, d i ! d for some nonzero d 2 T X (x).S i n c e X is convex and closed, for each i,w eh a v e

td i +x 2 X, 8t 2 [0,r], td +x 2 X, 8t 2 [0,r].
Thus, 0 6 = d 2 T X (x).S u p p o s et h a t{t i } is a sequence of scalars in [0,r] that converges to zero. By convexity of f and due to (3.17) and (3.18), we get

f (x + t i d i )  (1 -t i kx i -xk )f (x)+ t i kx i -xk f (x i )  f (x)+ t i kx i -xk C i (x -x i ).
Since C i ! 0,f r o mt h ec o n v e x i t yo ff and the above statement, we have ⇠ T d  0, with ⇠ being an n ⇥ p matrix whose jth column belongs to @f j (x). Therefore,

0 6 = d 2 G(x) \ T X (x)
. This gives a contradiction and completes the proof.

In the rest of this section, we consider a multi-objective optimization problem whose feasible set is defined by some constraint functions. Consider

min f (x) s.t. g i (x)  0,i =1, 2,...,m, (3.21)
where f : R n ! R p is the objective function (i.e., f (x)=(f 1 (x),...,f p (x)) and the g j functions define the constraints. Hereafter, whenever we use the Clarke subdifferential for the g j functions, we assume that these functions are locally Lipschitz.

For a feasible p oint x,t h ei n d e xs e tI(x) is defined by

I(x)={j 2{1, 2 ...,m} : g j (x)=0}.
In the following, we are going to provide a characterization of robust efficient solutions of Problem (3.21). The following constraint qualification (CQ) helps us in the sequel.

Definition 3.8. We say that constraint qualification (CQ) holds at x,i f

0 / 2 co ⇢ [ j2I(x)
@g j (x) .

Theorem 3.4. If x is a robust efficient solution which satisfies (CQ), then

pos ✓ p [ i=1 @f i (x) ◆ + pos ✓ [ i2I(x) @g i (x) ◆ = R n .
Proof. For simplicity, we set

I x = pos ✓ p [ i=1 @f i (x) ◆ + pos ✓ [ i2I(x) @g i (x) ◆ .
It can be seen that under the assumptions of the theorem and Theorem 1.8,

{d : g • i (x; d)  0, 8i 2 I(x)}✓T X (x).
Therefore, according to Theorem 3.2, the system below has no solution d 2 R n :

⇠ T d  0, 8⇠ 2 @f i (x), 8i 2{1,...,p} ⇠ T d  0, 8⇠ 2 @g i (x), 8i 2 I(x) d 6 =0.
Hence, the following system has no solution d 2 R n :

⇠ T d  0, 8⇠ 2 @f i (x), 8i 2{1,...,p} ⇠ T d  0, 8⇠ 2 @g i (x), 8i 2 I(x) d 1 > 0.
Using the semi-infinite Farkas Theorem 1.9, we have e 1 2 cl(I x). Similarly, it can be shown that ±e i 2 cl(I x) for each i 2{ 1, 2,...,p}.H e r e ,e i denotes the ith unit vector. Therefore, cl(I x)=R n . Since A x is a convex set whose closure is equal to R n , we have I x = R n and the proof is complete.

Corrolary 3.1. Assume that f i (i =1 ,...,p) and g j (j =1 ,...,m) in Problem

✓ p [ i=1 @f i (x) ◆ + pos ✓ [ i2A(x) @g i (x) ◆ = R n ,
then x is a robust efficient solution of Problem (3.21).

Proof. We prove the theorem by contradiction. Supp ose that x is not robust.

Then, according to Theorem 3.3, there exists a nonzero vector d 2 T X (x)\G(x).F r o m the convexity assumption, we get ⇠ T d  0,foreac h⇠ 2 pos(@g i (x)) and each i 2 I(x).

Also, ⇠ T d  0,f o re a c h⇠ 2 pos(@f i (x)) and each i 2{ 1, 2,...,p}, because of d 2

G(x).O nt h eo t h e rh a n d ,b yt h ea s s u m p t i o no ft h et h e o r e m , d =

p X i=1 u i ⇠ i + X j2I(x) v j ζ j ,
for some u i ,v j ≥ 0,ξ i 2 pos(∂f i (x)),a n dζ j 2 pos(∂g j (x)). Therefore, dT d  0.

Hence, we get d =0which gives a contradiction. Proof. Suppose that x is not a properly efficient solution. Therefore, there exist {x i }✓X,increasingsequence{M i } of positive real numbers, and k 2{1,...,p},suc h that M i -! +1,

f k (x i ) <f k (x) 8i, (3.22)
and

f k (x) -f k (x i ) f j (x i ) -f j (x)
>M i , for each j 2{1,...,p} with f j (x i ) >f j (x).

(3.23)

Define Q i = {j : f j (x i ) >f j (x)}.
This set is nonempty because x is efficient. Without loss of generality, by choosing an appropriate subsequence, Q i is a constant set for all i indices. So, we denote it by Q. Also, define the feasible set of Problem (3.21) by X = {x 2 R n : g j (x)  0,j=1 , 2,...,m}. Without loss of generality, we assume that the sequence { x i -x kx i -xk } converges to some nonzero vector d.S e t t i n gt i = min{ 1 i , kx i -xk} and d i = x i -x kx i -xk ,w eh a v et i # 0 and x + t i d i 2 X,a c c o r d i n gt ot h e convexity assumptions. Hence, d 2 T X (x). Due to the convexity assumption, we get

f j (x i ) ≥ f j (x)+ξ T (x i -x), 8ξ 2 ∂f j (x), 8j 2{1,...,p}\Q, ) ξ T (x i -x)  0, 8ξ 2 ∂f j (x), 8j 2{1,...,p}\Q, ) ξ T d  0, 8ξ 2 ∂f j (x), 8j 2{1,...,p}\Q.
Moreover, from (3.23) and the convexity of the objective functions, we have

ξ T (x i -x)  f j (x i ) -f j (x), 8ξ 2 ∂f j (x), 8j 2 Q, < f k (x)-f k (x i ) M i , 8ξ 2 ∂f j (x), 8j 2 Q,  1 M i η T (x -x i ), 8η 2 ∂f k (x).
Thus,

ξ T d  0, 8ξ 2 ∂f j (x), 8j 2 Q. Therefore, d 2 T X (x) \ G(x)
. This is a contradiction because of Theorem 3.2, and the proof is complete.

Remark 3.1. The robust solution studied in the present chapter may not exist in some special cases, though these solutions (if exist) have nice properties as compared to non-robust points. An efficient point is robust if it stays efficient under small linear perturbations. Let us assume that the f i and g j functions are differentiable here. Under some CQs and appropriate assumptions, the KKT/FJ condition

p X i=1 λ i rf i (x)+ X j2I(x) µ j rg j (x)=0
for some nonnegative µ j 's and some nonnegative λ i 's (not all zero), is necessary for Remark 3.2. The necessary condition presented in Theorem 3.2 provides a tie-in to the gradient-like descent methods existing in the literature for solving vector optimization problems; see [START_REF] Drummond | Ap r o j e c t e dg r a d i e n tm e t h o df o r vector optimization problems[END_REF] and [START_REF] Fliege | Steepest descent methods for multicriteria optimization[END_REF]. Extending these numerical tools to generate robust solution(s) can be worth studying in future.

Setting

d o = x o -x kx o -xk ,w eh a v ekd o k =1and d o 2 T X (x)
due to the convexity of X. Furthermore, by convexity of f ,w eg e t

f 0 (x; d o )+C o d o = f 0 (x; x o -x) kx o -xk + C o d o  f (x o ) -f (x) kx o -xk + C o (x o -x) kx o -xk .
Therefore, according to (3.24), we get

f 0 (x; d o )+C o d o 2-R p = . (3.25)
Defining

⇢ o =sup{t : f 0 (x; d o )+tCd o / 2-R p = , 8kCk1}, (3.26)
we have ⇢  ⇢ o .F u r t h e r m o r e ,f o re a c ht 2 (0,⇢ o ) and each C with kCk1,w eh a v e

f 0 (x; d o )+tCd o / 2-R p = .
This is a contradiction with (3.25) by setting t = kC o k and C = C o kC o k ,a n dt h ep r oo fi sc o m p l e t e . It can be seen that the optimal value of the optimization problem considered in the above theorem is equal to the maximum robustness radius, if one furthermore assumes the equality of the Bouligand tangent cone and the cone of feasible directions.

Comparison with other notions

In the following, we highlight the relationships between the robustness notion considered in this chapter and some notions mentioned in Section 3.1.2. See also [START_REF] Georgiev | Robust aspects of solutions in deterministic multiple objective linear programming[END_REF] for some comparisons. Proof. To the contrary assume that

f U (x o ) ✓ f U (x) -R p ≥ (3.27) for some x o 2 X. This implies 8C 2 U 9 C 2 U s.t. f (x o )+Cx o  f (x)+ C x.
(3.28)

The two vectors x o and x can not be zero. If x o =0,t h e nb y( 3 . 2 8 ) ,

f (0) + C(0)  f (x)+ C x,
or some C 2 U . This contradicts the robustness assumption. Moreover, if x =0 , then by considering C =0in (3.28), there exists some C with k Ck0.5✏ such that

f (x o )  f (x).
This contradicts the efficiency of x.H e n c e ,x 0 6 =0and x 6 =0.

Now, define

M = {λ 2 R p ≥ : kλk1, p X j=1 λ j ≥ 1}.
It is clear that M is a nonempty compact convex set. Let F : M ◆ M be a set-valued mapping defined by

F (λ)={λ 0 2 M : f (x o )+ ✏kx o k 2kλk λ  f (x)+ ✏kxk 2kλ 0 k λ 0 }.
We show that F (λ) is nonempty and convex for each λ 2 M .

Let λ 2 M .D e fi n i n gt h ep ⇥ n matrix C o := ✏ 2kλkkx o k λx o T .W eh a v ekC o k0.5✏, and hence by (3.28), there exists some p ⇥ n matrix C such that kCk0.5✏,a n d

f (x o )+ ✏kx o k 2kλk λ  f (x)+C x. (3.29)
Consider λ with λi = kC i k.D e fi n eλ 0 := λ k λk . By considering the Cauchy-Schwarz inequality and

✏ 2k λk ≥ 1,w eh a v e f (x o )+ ✏kx o k 2kλk λ  f (x)+C x  f (x)+kxk λ  f (x)+ ✏kxk 2k λk λ  f (x)+ ✏kxk 2kλ 0 k λ 0 Therefore, due to λ 0 2 M ,w eh a v eλ 0 2 F (λ),a n dh e n c eF (λ) is nonempty.
To prove the convexity, let λ 1 ,λ 2 2 F (λ) and υ 2 (0, 1).F i r s t ,w ea s s u m et h a t

kλ 1 k = kλ 2 k =1.
Then, by definition of F (λ),w eg e t

f (x o )+ ✏kx o k 2kλk λ  f (x)+ ✏kxk 2 (υλ 1 +(1-υ)λ 2 ) Due to kυλ 1 +(1-υ)λ 2 k1 and υλ 1 +(1-υ)λ 2 ≥ 0,w ec a ni n f e r f (x o )+ ✏kx o k 2kλk λ  f (x)+ ✏kxk 2kυλ 1 +(1-υ)λ 2 k (υλ 1 +(1-υ)λ 2 ).
Hence,

υλ 1 +(1-υ)λ 2 2 F (λ) when kλ 1 k = kλ 2 k =1. Now, considering two arbitrary vectors λ 1 ,λ 2 2 F (λ)
and υ 2 (0, 1),t h e r ea r eγ>0 and µ 2 (0, 1) such that

υλ 1 +(1-υ)λ 2 = γ(µ λ 1 kλ 1 k +(1-µ) λ 2 kλ 2 k ). (3.30)
Notice that 0 < kλ 1 k, kλ 2 k1.B yd e fi n i t i o no fF (λ),i ti sc l e a rt h a t λ 1 kλ 1 k , λ 2 kλ 2 k 2 F (λ).F u r t h e r m o r e ,i fλ 0 2 F (λ) and γλ 0 2 M ,f o rs o m eγ>0,t h e nγλ 0 2 F (λ).

Therefore, according to (3.30), we have

υλ 1 +(1-υ)λ 2 2 F (λ). Hence, F is convex- valued.
It is clear that graph of F is closed. Therefore, by the Kakutani fixed-point Theorem 1.4, there exists some λ ⇤ 2 M such that

f (x o )+ ✏kx o k 2kλ ⇤ k λ ⇤  f (x)+ ✏kxk 2kλ ⇤ k λ ⇤ . (3.31)
The above inequality does not hold as equality, since otherwise due to (3.28) we have

f (x)+ ✏kxk 2kλ ⇤ k λ ⇤ = f (x o )+ ✏kx o k 2kλ ⇤ k λ ⇤  f (x)+ e C x,
for some e C with k e Ck0.5✏.

Then ✏kxk 2kλ ⇤ k λ ⇤  e C x.
By the Cauchy-Schwarz inequality,

we get ✏kxk 2kλ ⇤ k λ ⇤ kxkd in which d 2 R p with d i = k e C i k. Therefore, k e Ck = kdk > 0.5✏
which is a contradiction. Thus, inequality (3.31) holds and it does not hold as equality.

On the other hand, by Cauchy-Schwarz inequality,

f (x o )+ ✏x T x o 2kλ ⇤ kkxk λ ⇤  f (x o )+ ✏x o T x o 2kλ ⇤ kkx o k λ ⇤ .
Hence, according to (3.31),

f (x o )+ ✏x T x o 2kλ ⇤ kkxk λ ⇤  f (x)+ ✏x T x 2kλ ⇤ kkxk λ ⇤ . Therefore, setting C o = ✏ 2kλ ⇤ kkxk λ ⇤ xT ,w eh a v ekC o k <✏and f (x o )+C o x o  f (x)+C o x.
The two last relations contradict the robustness of x (in the sense of Definition 3.6)

and the proof of the first part is complete.

As co(f U (x)) = f U (x), x is also hull-based minmax robust efficient.

It is not difficult to see that Theorem 3.8 will be valid if one replaces 0.5✏,i nt h e considered uncertainty set, with any ✏ 2 (0,✏).N o t et h a tt h ec o n v e r s eo fn o n eo ft h e propositions given in this section holds, in general.

Modification of the objective function

In this section, we consider two robustness aspects of (weakly/properly) efficient solutions. In the first one, we consider a convex combination of the objective function of Problem (3.21) with a new special function. The second robustness aspect is due to adding a new objective function to the problem. In both cases, we examine preserving the weak/proper/robust efficiency.

Consider the following problem for ↵ 2 [0, 1]:

min f (x)+(1-↵)h(x)q s.t. g i (x)  0,i 2{1, 2,...,m},
where h : R n ! R is a convex function and q 2 R p ≥ is a p-vector with nonnegative components. We denote this program by (MOP) ↵ , and this program coincides with (3.21) when ↵ =1.

Note: Throughout this section, we assume that the functions h, f i (i =1,...,p) and g j (j =1,...,m) are convex and there is x such that g j (x) < 0( j =1,...,m). Theorem 3.9 presents a sufficient condition for properly efficient solutions of problems (3.21) and (MOP) 0 to remain properly efficient for (MOP) ↵ . Theorem 3.9. If x is a properly efficient solution of both problems (3.21) and (MOP) 0 ,t h e nx is a properly efficient solution of (MOP) ↵ for each ↵ 2 (0, 1).

Proof. Since x is a properly efficient solution of Problem (3.21), then there exist

λ 2 R p and w 2 R m such that 0 2 p X i=1 λ i @f i (x)+ m X j=1
w j @g j (x),w j g j (x)=0,j =1,...,m, λ > 0,w ≥ 0. Also, since x is a properly efficient solution of Problem (MOP) 0 ,t h e r ee x i s tµ 2 R p and v 2 R m such that µ>0,v≥ 0,a n d

0 2 p X i=1 µ i @f i (x)+q T µ@h(x)+ m X j=1 v j @g j (x), (3.32)
v j g j (x)=0,j =1,...,m.

(3.33)

Notice that the convexity of h is crucial in obtaining (3.32).

Let ↵ 2 (0, 1).W ed e fi n et and γ as follows:

t := ↵q T µ ↵q T µ+(1-↵)q T λ , γ := tλ +(1-t)µ.
It is clear that 0 <t<1 and γ>0.A l s o ,

(1 -t)q T µ =(1-↵)q T γ. (3.34) Thus, p X i=1 γ i @f i (x)+(1-↵)q T γ@h(x)=t p X i=1 λ i @f i (x)+(1-t) p X i=1 µ i @f i (x)+(1-t)q T µ@h(x).
Therefore, setting z = tw +(1-t)v,w eg e t 0 2 P p i=1 (tλ i +(1-t)µ i )@f i (x)+(1-t)q T µ@h(x)+ P m j=1 (tw j +(1-t)v j )@g j (x) = P p i=1 γ i @f i (x)+(1-↵)q T γ@h(x)+ P m j=1 z j @g j (x),

where γ>0 and z ≥ 0. Therefore, x is a global minimizer for

min P p i=1 γ i f i (x)+(1-↵)q T γh(x) s.t. g j (x)  0,j =1,...,m.
This implies that x is a properly efficient solution of (MOP) ↵ , according to Theorem 3.11 of [START_REF] Ehrgott | Multicriteria optimization[END_REF].

The following two results give sufficient conditions for efficient (respectively weakly) solutions of problems (3.21) and (MOP) 0 to remain efficient (resp. weakly) for (MOP) ↵ . These results extend Proposition 2.2 of [START_REF] Georgiev | Robust aspects of solutions in deterministic multiple objective linear programming[END_REF]. Theorem 3.10. Let x be an efficient solution of both Problems (3.21) and (MOP) 0 .

Then x is an efficient solution of (MOP) ↵ for each ↵ 2 (0, 1).

Proof. Let ↵ 2 (0, 1).T ot h ec o n t r a r ya s s u m et h a tt h e r ee x i s t saf e a s i b l ep o i n t ,

x,s u c ht h a t

f (x)+(1-↵)qh(x)  f (x)+(1-↵)qh(x), f (x)+(1-↵)qh(x) 6 = f (x)+(1-↵)qh(x). If h(x) <h(x),t h e n f (x) -f (x)  (1 -↵)q(h(x) -h(x))  (and 6 =) 0.
This contradicts the efficiency of x for (3.21). Hence, we assume h(x) ≥ h(x).D u e to the convexity assumption, we have

f ( 1 2 x + 1 2 x)+qh( 1 2 x + 1 2 x)  1 2 f (x)+ 1 2 f (x)+ 1 2 qh(x)+ 1 2 qh(x)  (and 6 =) f (x)+ 1 2 (1 -↵)q ✓ h(x) -h(x) ◆ + 1 2 qh(x)+ 1 2 qh(x) = f (x)+q ✓ h(x)+ ↵ 2 h(x) -h(x) ◆  f (x)+qh(x).
Hence, setting z = 1 2 x + 1 2 x,t h ev e c t o rz is feasible and

f (z)+qh(z)  f (x)+qh(x).
This contradicts the efficiency of x for (MOP) 0 ,a n dc o m p l e t e st h ep r oo f . Theorem 3.11. Let x be a weakly efficient solution of both Problems (3.21) and

(MOP) 0 .T h e nx is a weakly efficient solution of (MOP) ↵ for each ↵ 2 (0, 1).

Proof. The proof is similar to that of Theorem 3.10 and is hence omitted.

@f i (x) ◆ + pos ✓ [ i2A(x) @g i (x) ◆ = R n and pos ✓ p [ i=1 @ f i + q i h (x) ◆ + pos ✓ [ i2I(x) @g i (x) ◆ = R n .
By the above two equalities, and since all the @-sets are convex here, we have pos

✓ p [ i=1 @ f i +(1-↵)q i h (x) ◆ + pos ✓ [ i2I(x) @g i (x) ◆ = R n .
Therefore, x is a robust efficient solution for (MOP) ↵ , because of Theorem 3.5.

In the rest of this section, we examine adding a new objective function to Problem (3.21). Consider the following multi-objective optimization problem, denoted by

(MOPh): min 0 @ f (x) h(x) 1 A s.t. g i (x)  0 i =1,...,m,
where h : R n ! R. The following two theorems address some connections between the properly efficient solutions of the two problems (3.21) and (MOPh).R e c a l lt h a t the functions h, f i ,a n dg j are convex.

Theorem 3.13. Let x be a properly efficient solution of Problem (MOPh).I f

@h(x) ✓ pos ✓ p [ i=1 @f i (x) ◆ + pos ✓ [ i2I(x) @g i (x)
◆ , then x is a properly efficient solution of Problem (3.21).

Proof. Since x is a properly efficient solution of Problem (MOPh),t h e nt h e r e exist (λ, λ p+1 ) 2 R p ⇥ R and w 2 R m such that 0 2 P p i=1 λ i rf i (x)+λ p+1 @h(x)+ P m j=1 w j rg j (x), w j g j (x)=0,j=1,...,m, λ > 0,w ≥ 0.

Therefore, there exists some d 2 @h(x) such that

d = - P p i=1 λ i λ p+1 rf i (x) - P m j=1 w j λ p+1 rg j (x).
On other hand, x is a properly efficient solution of Problem (3.21). Therefore, there exist λ 0 2 R p and w 0 2 R m such that

0= P p i=1 λ 0 i rf i (x)+ P m j=1 w 0 j rg j (x), (3.35)
w 0 j g j (x)=0,j=1,...,m λ 0 > 0,w 0 ≥ 0.

(3.36) Let t>max 1ip { λ i λ 0 i λ p+1 }.W eh a v e d = p X i=1 (tλ 0 i - λ i λ p+1 )rf i (x)+ m X j=1 (tw 0 j - w j λ p+1 )rg j (x).
Setting u i := tλ 0 i -λ i λ p+1 and v j = tw 0 j -

0=

P p i=1 (tλ 0 iu i )rf i (x)+d + P m j=1 (tw 0 j + µ j )rg j (x) 2 P p i=1 (tλ 0 iu i )rf i (x)+@h(x)+ P m j=1 (tw j + µ j )rg j (x).

Therefore, x is a properly efficient solution of (MOPh),andtheproofiscomplete.

The following example shows that part (i) of the above theorem may not hold when some f i or g j functions are nonsmooth. A similar example can be constructed for part (ii).

Example 3.5. Let f : R ! R be defined by

f (x)= 8 < : x 2 -x 1 ,x 2 > 0 -x 1 ,x 2  0.
Consider the following optimization problem:

min f (x) s.t. g(x)=x 1 -x 2  0.
The functions f and g are convex and

@f 0 @ 0 0 1 A = 8 < : 0 @ -1 ↵ 1 A : ↵ 2 [0, 1] 9 = ; ,@ g 0 @ 0 0 1 A = 8 < : 0 @ 1 - 1 1 A 9 =
;

.

With λ = ↵ = µ =1,w eh a v e 0 @ 0 0 1 A = λ 0 @ -1 ↵ 1 A + µ 0 @ 1 -1 1 A . Therefore, x = 0 @ 0 0 1
A is an optimal solution of the above problem. Now, consider the function h(x)=x 1 and the following problem

min 0 @ f (x) h(x) 1 A s.t. g(x)=x 1 -x 2  0.
(3.37)

We have @h(x)=

1 A = λ 0 @ -1 ↵ 1 A + µ 0 @ 1 -1 1 A ,
for some µ 2 R and ↵ 2 [0, 1]. It shows that part (i) of Theorem 3.14 may not be valid in the presence of nonsmooth f i or g j functions.

The last theorem of this section establishes a connection between the robust solutions of two problems (3.21) and (MOPh).

✓ p [ i=1 @f i (x) ◆ + pos ✓ [ i2I(x) @g i (x) ◆ .
Proof. The results follow from Theorems 3.3 and 3.6.

Chapter 4

Stability and sensitivity analysis

Introduction

This chapter is concerned with sensitivity analysis and stability in parametrized multiobjective optimization. A fundamental question in optimization is the investigation of behavior of solution set and efficient values under perturbation. Considerable work has been devoted to these topics under the titles of stability and sensitivity analysis. The stability term is used for examination of continuity of optimal value and solution set under perturbation. By sensitivity analysis we mean studying differentiability of the optimal value function and solution set. Moreover, this served as an important motivation for proposing some generalized derivatives including proximal subdifferential.

Here, we only mention some works dedicated to sensitivity analysis and stability in multi-objective optimization. We refer the reader to [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF] and references therein for single objective optimization.

Stability in optimization problems is both theoretically and practically important.

It has been used as a useful tool in post-optimal analysis. There are many papers addressing stability in vector optimization. Some scholars have considered parametric vector optimization with the objective function and the feasible set depending on a parameter lying in a Banach space. They provide some sufficient conditions for lower and upper semicontinuity of set-valued mappings corresponding to feasible set, for their derivatives. We also provide some sufficient conditions for their pseudo-Lipschitz continuity. Thanks to the mentioned theorem, we provide sufficient conditions for semi-differentiability and pseudo-Lipschitz continuity of efficient solutions and efficient values. Furthermore, some formulas for their derivatives are given. In addition, some counterexamples are given to clarify the theoretical results.

The rest of the chapter unfolds as follows. Section 2 is devoted to strict semidifferentiability and pseudo-Lipschitz continuity of the set-valued mappings of feasible set and feasible values. We establish that under mild conditions, the set-valued mapping of feasible set and its feasible value are strictly semi-differentiable. Moreover, we investigate the pseudo-Lipschitz continuity of the mentioned set-valued mappings. In Section 3, we provide some sufficient conditions for pseudo-Lipschitz continuity and semi-differentiability of the efficient solutions and efficient values.

Parametric multi-objective optimization

Consider the following parametric multi-objective optimization problem, denoted as

P (u), min f (u, x) s.t. g(u, x) 5 0, (4.1)
where f : R n ⇥ R q ! R p and g : R n ⇥ R q ! R m are locally Lipschitz functions.

Here, x is a decision vector and u is a parameter. Associated with this problem, the following set-valued mappings, called feasible solution, feasible value, efficient solution and efficient value mappings, are respectively defined by

X(u):={x 2 R n : g(u, x) 5 0} Y (u):={f (u, x):x 2 X(u)} E(u):={x 2 X(u):x is an efficient solution of P (u)} V (u):={f (u, x):x 2 E(u)}
In the sequel, we say that constraint qualification CQ holds at (ū, x) 2 gphX if 0 / 2 co([ i2I(ū,x) @ x g i (ū, x))

where I(ū, x) denotes the index set of active constraints at (ū, x) and @ x stands for generalized gradient with respect to x. It is well-known that this constraint qualification is equivalent to

M (R n )+R |I(ū,x)| = = R |I(ū,x)| , 8M 2 @ x g I(ū,x) ,
where g I(ū,x) is the function defined by the active components of g at (ū, x) and |I(ū, x)| is equal to the cardinal number of I(ū, x). The next remark provides a sufficient condition for pseudo-Lipschitz continuity of X at a given point [START_REF] Rockafellar | Lipschitzian properties of multifunctions[END_REF]. Proof. Since CQ holds at (ū, x) and g is regular at this point, gphX is regular at (ū, x); see Theorem 1.8. Therefore, X is Clarke differentiable at the given point.

Strict semi-differentiability follows from Proposition 1.2 and Remark 4.1. Due to the mentioned facts, derivative of X at (ū, x) is given by the above formula.

The following examples illustrate that all the given assumptions of Proposition which shows that d 2{ v : f 1 X (u; v)=0 }. This contradicts the hypothesis because d 2 X 1 (ū) and d 6 =0.H e n c e ,{x n } admits some cluster points. Moreover, on account of continuity of f and g,c l u s t e rp o i n t sb e l o n gt o X(ū, ȳ). Now, the result can be obtained similar to Theorem 4.1.

Example 4.3. Let X(u)={x 2 R :0 x  p |u|} [ [2, 3] and f (u, x)= 8 > < > : x, x  1 2 -x, x ≥ 1.
It can be verified easily that neither (i) nor (ii) holds at (ū, ȳ)=( 0 , 0) and is not pseudo-Lipschitz continuous at the point.

In the next theorem, we provide some sufficient conditions for strict semi-differentiability of Y .

Theorem 4.3. Let y 2 Y (ū) be given and let gphY be closed around (ū, ȳ).I ft h e following conditions hold, then Y is strictly semi-differentiable at (ū, ȳ).

(i) there exists x 2 X(ū, ȳ) such that for each sequence {(u n ,y n )}✓gphY with (u n ,y n ) ! (ū, ȳ), x is a cluster point for some sequence {x n } with {(u n ,y n ,x n )}✓ gph X;

(ii) X is pseudo-Lipschitz continuous and strictly semi-differentiable at (ū, x) and f is regular at this point;

(iii) DX(ū, x)(0) \ Kerf 0 x (ū, x)={0},
where Kerf 0 x (ū, x)={d : f 0 (ū, x;0,d)=0}. In addition,

D s-low Y (ū, ȳ)(l)={f 0 (ū, x; l, d):8d 2 D s-low X(ū, x)(l)}, 8l 2 R q . (4.3)
Proof. First, we show that if (l, e) 2 T gphY ,thene 2{f 0 (ū, x; l, d):8d 2 D s-low X(ū, x)(l)}.

Let (l, e) 2 T gphY .B yt h ed e fi n i t i o no fB o u l i g a n dt a n g e n tc o n e ,t h e r ee x i s t{(u n ,y n )}✓

(iii) DX(ū, x)(0) \ Kerf 0 x (ū, x)={0},f o re v e r yx 2 X(ū, ȳ).
Then Y is strictly semi-differentiable at (ū, ȳ) and for each l 2 R q D s-low Y (ū, ȳ)(l)={f 0 (ū, x; l, d):8x 2 X(ū, ȳ), 8d 2 D s-low X(ū, x)(l)}.

Proof. The proof is similar to that of Theorem 4.

4.4. Let X(u)={(x 1 ,x 2 ):x 1 + x 2 =1,x 1 ,x 2 ≥ 0} and f (u, x)=u(x 1 - x 2 ).A t(ū, ȳ)=(0, 0),w eh a v e T gphY (ū, ȳ)=pos{(1, 1), (1, -1)}[pos{(-1, 1), (-1, -1)}, D s-low Y (ū, ȳ)(l)={0},
for each l.T h u s ,Y is not strictly semi-differentiable at (ū, ȳ).

In the next lemma we provide a sufficient condition for having the hypothesis of Remark 4.2.

with k being a positive constant independent of the sequence {(u n ,y n )}.

Proof. This follows from Theorem 1.7.

We say CQ1 holds at (ū, x) if we have assumptions of Lemma 4.1.

Since efficient points of the two sets Y (u) and Y (u)+R p = are the same, it would be in some cases beneficial to consider the set-valued mapping Y + R p = , where Y + R p = (u)=Y (u)+R p = . For instance, when the functions g and f are convex with respect to x, Y (u) may not be convex while Y (u)+R p = is convex. So, we are equipped with the great tools of convex analysis. Another noticeable case is when

Y (u) is not closed while Y (u)+R p = is closed.
In what follows we investigate pseudo-Lipschitz continuity and strict semi-differentiability of this set-valued mapping. (i) Y is pseudo-Lipschitz continuous at (ū, ȳ) and gphY and gph(Y +R p = ) are closed around it;

(ii) Y 1 (ū) \-R p = = {0}.
Proof. We show that if (σ, 0) 2 N gphY +R p = (ū, ȳ) then σ =0.O nt h ec o n t r a r y ,s u p p o s e that σ 6 =0.B yt h ed e fi n i t i o no fl i m i t i n gc o n e ,t h e r ea r es e q u e n c e s{(u n ,y n + r n )}✓ gphY + R p = and {(σ n ,µ n )} such that (u n ,y n + r n ) ! (ū, ȳ) and (σ n ,µ n ) ! (σ, 0). Note that {r n } is bounded, since If this is not the case, then we would assume without loss of generality that r n !1and kr n k -1 r n ! r, where r is a nonzero vector in R p = . Therefore,

lim n!1 (u n -ū, y n kr n k )=(0, -r),
which contradicts our assumptions. Moreover, r n ! 0,s i n c ei ft h i sd o e sn o th o l d , then it leads to a contradiction with ȳ 2 V (ū).A s µ n ! 0,t h e r ei sas e q u e n c e

{✏ n }✓R > such that hσ n ,u-u n io(ku-u n k+kz-y n -r n )k)+✏ n (ku-u n k+kz-y n -r n )k), 8(u, z) 2 gphY +R p = ,
and ✏ n # 0. Moreover, clearly we have

hσ n ,u-u n io(ku -u n k + ky -y n )k)+✏ n (ku -u n k + ky -y n )k), 8(u, y) 2 gphY, which implies σ 2 N gphY (ū, ȳ) (Theorem 1.2), contradicting pseudo-Lipschitz conti- nuity of Y at (ū, ȳ).
The contingent differentiability and the Proto differentiability of Y +R p = have been investigated in [START_REF] Lee | On sensitivity analysis in vector optimization[END_REF][START_REF] Tanino | Sensitivity analysis in mcdm[END_REF]. In the following theorem, we present sufficient conditions for strict semi-differentiability of Y + R p = . (i) Y is strictly semi-differentiable at (ū, ȳ) and Y is closed at ū;

(ii) Y 1 (ū) \-R p = = {0}.
Then, Y + R p = is strictly semi-differentiable at (ū, ȳ) and The property Y 1 (ū) \-R p = = {0} plays an important role here. It is readily seen that if Y is bounded from below on a neighborhood of ū then this property holds.

D s-low (Y + R p = )(ū, ȳ)(l)=D s-low Y (ū, ȳ)(l)+R p = 8l 2 R q . ( 4
In the following propositions, we provide some sufficient conditions for having this property. Proof. Suppose, contrary to our claim, that there is a nonzero vector e 2 Y 1 (ū) \ -R p = .B yd e fi n i t i o n ,t h e r ea r es e q u e n c e s{(u n ,y n )}✓gphY and {t n }✓R > such that (u n ,t n y n ,t n ) ! (ū, e, 0). Y being pseudo-Lipschitz at (ū, ȳ),t h e r ei sas e q u e n c e {(u n ,z n )}✓gphY with z n ! ȳ. Thus, for n sufficiently large t n y n +(1-t n )z n 2 Y (u n ). This implies ȳr 2 Y (ū) which contradicts efficiency of ȳ. Thus, e =0and the proof is complete.

Shi [START_REF] Shi | Contingent derivative of the perturbation map in multiobjective optimization[END_REF] and Lee [START_REF] Lee | On sensitivity analysis in vector optimization[END_REF] obtained formula (4.5) for contingent derivative and Proto derivative, respectively, under the assumption of D s Y (ū, ȳ)(0)\-R p = = {0} instead of hypotheses (ii). In the following proposition, we show that this hypothesis is satisfied when we have D s Y (ū, ȳ)(0) \-R p = = {0}.

z n  y n and kz n k≥n + ky n k 2 .L e tu se x t r a c tas u b s e q u e n c e{kz n k -1 z n } without relabeling that converges to some nonzero vector r. Therefore,

lim n!1
(u n , kz n k -1 z n )=(0,r).

Since z n  y n , r 2-R p = which contradicts our assumptions. In virtue of pseudo-Lipschitz continuity of Y , Y is nonempty in some neighborhood of ū. This implies non-emptiness of V in a neighborhood of ū. Y is locally dominated around ū (Lemma 4.2), and so we can choose a sequence {r n }✓R p = such that y nr n 2 V (u n ) for n sufficiently large. Note that {r n } tends to zero, since if it were not the case, either {r n } has a convergent subsequence tending to a nonzero vector or it tends to infinity. In the first case, because y n ! ȳ,w eobtain ŷ 2 Y (ū) with ŷ  ȳ. This is a contradiction with ȳ 2 V (ū).I nt h es e c o n dc a s e , reducing to subsequence if necessary, lim n!1 (u n , y nr n kr n k =(0, -r),

where r 2 R p ≥ . This is a contradiction with condition (iii). We can infer from (4.7) that for each δ>0 and for n sufficiently large, we have ✏(ku nūk + ky nȳk) ku nūk + ky nr nȳk  hσ, u nūi ku nūk + ky nr nȳk  δ.

The above inequality and the triangle inequality imply lim n!1

(u nū, y nȳ) kr n k =(0, 0).

Without loss of generality, we assume that {kr n k -1 r n } converges to a nonzero vector r 2 R p ≥ . Thus, we have and (u i n ,y i n ) ! (u n ,y n ).I fi ti sn o tt h ec a s e ,t h e nw eo b t a i nac o n t r a d i c t i o n(σ, 0) 2 N gphY (ū, ȳ).S i m i l a r t o t h e fi r s t c a s e , f o r s u ffi c i e n t l y l a r g e n,w eo b t a i nav e c t o r r 2-R p ≥ such that r 2 DY (u n ,y n )(0). This leads to a contradiction with the assumptions, and the proof is complete.

In the following propositions, we provide a lower and upper estimation of D low V (ū, ȳ). Proof. this can be proven similar to Theorem 3.3. of [START_REF] Kuk | Sensitivity analysis in vector optimization[END_REF]. )=(l, h) which is impossible. Therefore, {h n } tends to e, which is the desired result.

The following example shows that the upper and lower estimations of D low V (ū, ȳ)(l) can be strict. The next theorem investigates the semi-differentiability of V at a given point.

Theorem 4.9. Let the fol lowing conditions hold:

(i) Y is semi-differentiable at (ū, ȳ) and it is locally dominated around ū;

(ii) DY (ū,ȳ) (0) \-R p = = {0};

(iii) Y 1 (ū) \-R p = = {0}.

If ȳ 2 V (ū) is a uniformly proper efficient point, then V is semi-differentiable at where K ✏ = \ p i=1 {d : he i + ✏e, di≥0}, e i is the unit vector of the standard basis and e is a vector with all components being to one. It is clear that if for some convex, closed and pointed cone R p = ✓ int(K) [{0},t h e nt h e r ei s✏ such that K ✏ ✓ K. Moreover, by using Farkas' lemma, if we have

8 > < > : Y (u n ) \ (y n -R p = )={y n } {0} * (Y (u n ) -y n ) \-K ✏ ,
then there is λ n = 0 such that -λ n 2 N Y (un) (y n ), P p i=1 λ i n =1and min 1ip λ i n  ✏. We consider a sequence with ✏ n # 0. By Cantor's diagonal argument and the aforementioned fact, if necessary reducing to a subsequence, we can construct sequence {(u n ,y n )}✓gphV and {λ n } such that for some j 2{1,...,p} The following example shows that we can have uniformly proper efficiency at a given point while not having normally proper.

  and X PE are not necessarily closed. Also, setting Y = f (X), the set of nondominated points, denoted by Y N ,i sd e fi n e db yY N = f (X E ),a n dt h e set of properly nondominated points, denoted by Y PN ,i sd e fi n e db yY PN = f (X PE ).

  at x =0.H o w e v e r ,i ff o rs o m en e i g h b o r h o o dO of x and each x 2 O, Γ(x) is compact and if Γ is closed at x,t h e ni ti su p pe rs e m i c o n t i n u o u sa tt h epo i n t ;s e e[ 5 4 ] .Lower semicontinuity and upper semicontinuity are equivalent to continuity when Γ is single-valued. Of course they are distinct when being set-valued; see Figure1-3depicting two set-valued mappings from R to the power set of R.

Figure 1 - 3 -

 13 Figure 1-3 -Upper and lower semicontinuity of set-valued mappings

Theorem 1 . 4 .

 14 (Kakutani fixed-point theorem) Let Γ:C ◆ C be a given set-valued mapping. Suppose that C is compact and convex and Γ(x) is convex and nonempty for each x 2 C.I fg r a p ho fΓ is closed, then there exists x 2 C satisfying x 2 C(x).

Proposition 1 . 3 .

 13 Let f : R n ! R p be a local ly Lipschitz function with modulus k near x.

.

  N o n e t h e l e s s , equality does not hold necessarily; for instance, consider f (x)=

Question 1 . 1 .

 11 r e l a t e d question to the foregoing theorem is the following amazing open problem. Let f : R (n+p) ! R p be local ly Lipschitz. Suppose that for every M 2 @f(x), M has full rank. Is there some n ⇥ (n + p) matrix, N ,s u c ht h a t

Now, to the contrary, assume that ȳ/ 2 Y

 2 PN . Thus, according to Benson's proper efficiency definition, there are sequences {β k }✓(0, +1), {y k }✓Y and {d k }✓R p

  2{1,...,p}\Q. (3.9) Inequalities (3.8) and (3.9) contradict the robustness of x. Now, we consider the latter case, x =x.W ea s s u m et h a tt h es e q u e n c e{ x i -x kx i -xk } converges to some nonzero vector d.W ec h o o s e Cp⇥n satisfying k Cp⇥n k <✏and Cj d<-2δ, 8j 2 Q, (3.10) Cj =0, 8j 2{1,...,p}\Q, (3.11)

3.1 is essential. Example 3 . 2 .Theorem 3 . 2 .

 3232 Consider the multi-objective optimization problem, min (-x, x 3 )s.t. x 2 R.It is not difficult to see that x =1is a robust efficient solution (consider ✏ =0 .1), while the problem does not have any properly efficient solution. Now, we are going to provide a characterization of robust efficient solutions with respect to the non-ascent directions of the objective function and the Bouligand tangent cone of the feasible set.Definition 3.7. The vector d 2 R n is called a non-ascent direction of f at x if d T ⇠  0, for each ⇠ 2 @f i (x) and each i 2{ 1, 2,...,p}. G(x) denotes the set of all non-ascent directions of f at x.The following theorem presents a necessary condition for robustness. If x is a robust efficient solution of Problem (3.3), then T X (x)\G(x)= {0}.

( 3 . 4 )

 34 for any matrix C p⇥n , with kCk <✏.W ec h o o s eam a t r i x Cp⇥n such that k Ck <✏ and C,d < -2δe (3.13)

. 18 )

 18 Two cases may o ccur for the sequence {x i }. Either it has a subsequence convergent to x or it does not have any subsequence convergent to x.W ec o n s i d e rt h e s et w o possible cases and we get a contradiction in each case.

( 3 . 5 extend

 35 21) are continuously differentiable. If x is a robust efficient solution of Problem (3.21) which satisfies (CQ), then pos{rf 1 (x),...,rf p (x)} + pos{rg i (x):i 2 I(x)} = R n . Theorem 3.5 provides a converse version of Theorem 3.4. Theorems 3.4 and 3.Theorem 3.4 of Georgiev et al. [32]. Theorem 3.5. Let f i (i =1 ,...,p) and g j (j =1 ,...,m) in Problem (3.21) be convex. If x is an efficient solution and pos

Corrolary 3 . 2 .Theorem 3 . 6 .

 3236 Assume that f i (i =1 ,...,p) and g j (j =1 ,...,m) in Problem (3.21) are differentiable and convex. If x is an efficient solution and pos{rf 1 (x),...,rf p (x)} + pos{rg i (x):i 2 I(x)} = R n , then x is a robust efficient solution of Problem (3.21). Although the compactness assumption is essential in Theorem 3.1 (see Example 3.2), the following result shows that Theorem 3.1 remains valid without compactness of the feasible set for convex programming problems. Let f i (i =1, 2,...,p) and g j (j =1, 2,...,m) be convex in Problem (3.21). If x is a robust efficient solution of Problem (3.21), then x is a properly efficient solution of Problem (3.21).

  the efficiency of x.I fs o m eo b j e c t i v ef u n c t i o n ,s a yf 1 ,i sp e r t u r b e d ,t h e nrf 1 (x) is alerted and hence to preserve the KKT/FJ condition (efficiency of x), the Lagrange multiplier(s) of some other objective function(s) or some constraint function(s) should be changed. Hence, at least one other objective function or at least one constraint function is required for robustness, i.e., m + p ≥ 2.T h u s ,t h e r ei sn or o b u s ts o l u t i o n for unconstrained single objective problems. To show this analytically, let x be an arbitrary optimal solution of min x2R n h(x),w h e r eh : R n -! R.T h e n rh(x)=0 which implies rh(x)+C 6 =0for each C 6 =0 .T h e r e f o r e ,x is not optimal for min x2R n h(x)+Cx for each C 6 =0 .H e n c e ,x is not robust for min x2R n h(x).T h u s ,t h i s unconstrained problem does not have any robust solution. Now, consider an unconstrained multi-objective programming problem min x2R n f (x), with f : R n -! R p and p ≥ 2.H e r e ,m =0 .I fx is a robust solution, then by Corollary 3.1, pos{rf 1 (x),...,rf p (x)} = R n , and hence p ≥ n +1. For constrained problem (3.21) satisfying the assumptions of Corol lary 3.1, if x is a robust solution, then p + m ≥ n +1.T h i si sn o tr e s t r i c t i v ef o rp r a c t i c a lc a s e s , because in practice the problem has at least 2n constraints due to the lower and upper bounds on variables.

Proposition 3 . 4 .Theorem 3 . 8 .

 3438 Let x be a robust solution of Problem (3.3) with radius ✏.T h e n , x is flimsy and highly robust efficient for the following uncertain multi-objective optimization problem, min f (C, x) s.t. x 2 X, C 2 U, minmax robust efficiency. Let x be a robust solution of Problem (3.3) with radius ✏.T h e nx is as e t -b a s e dm i n m a xr o b u s ts o l u t i o no ft h ef o ll o w i n gu n c e r t a i np r o b l e m : min f (C, x) s.t. x 2 X, C 2 U, with uncertain set U = {C p⇥n : kCk < 0.5✏} and uncertain objective f (C, x)= f (x)+Cx.I na d d i t i o n ,x is also hull-based minmax robust efficient.

w jλ

  p+1 ,c o m p l e t e st h ep r oo fo fp a r t( i ) . (ii) Setting µ j =0 ,f o re a c hj/ 2 I(x),b yt h ea s s u m p t i o no ft h et h e o r e m ,w eh a v e 0=-P p i=1 u i rf i (x)+d + P m j=1 µ j rg j (x),f o rs o m ed 2 @h(x). On other hand, since x is a properly efficient solution of Problem (3.21), there exist λ 0 2 R p and w 0 2 R m satisfying (3.35) and (3.36). For t>max 1ip { u i λ 0 i },w eh a v e

1 A 1 A

 11 s o ,f o rλ 1 = λ 2 =1and ↵ = µ =0,w eg e t is a properly efficient solution of Problem (3.37). Hence, in this example, is a properly efficient solution of both problems (3.21) and (MOPh),w h i l et h e r ei sn o ta n yλ>0 satisfying

Theorem 3 . 15 .

 315 If x be a robust efficient solution of Problem (3.21), then x is a robust efficient solution of (MOPh).T h ec o n v e r s eh o l d si f @h(x) ✓ pos

Remark 4 . 1 .Proposition 4 . 1 .

 4141 Let (ū, x) 2 gphX.I fC Qh o l d sa t(ū, x),t h e nX is pseudo-Lipschitz continuous at (ū, x). Let (ū, x) 2 gphX and let g be regular at this point. If CQ holds at (ū, x),t h e nX is strictly semi-differentiable at (ū, x) and for each l 2 R q D s-low X(ū, x)(l)={d : g 0 i (ū, x; l, d)  0, 8i 2 I(ū, x)}. (4.2)

4. 1

 1 are essential.

Example 4 . 1 .

 41 Let g(u, x)=x -max{u, 1}.I tc a nb ev e r i fi e de a s i l ya t(ū, x)=(1, 1) that DX(ū, x)(1) = {l : -1 <l 1},D s-low X(ū, x)(1) = {l : -1 <l 0}.

3 and Theorem 4. 2 . 4 . 2 .

 242 Remark The hypothesis DX(ū, x)(0) \ Kerf 0x (ū, x)={0} in Theorem 4.3 can be replaced by " (i) holds in Theorem 4.1 and for some positive constant kkx n -xkkku nūk + kky nȳk."This statement can be established in a similar way.The following example shows that the hypotheses assumed in Theorems 4.3, Theorems 4.4 and Remark 4.2 are essential for deriving strict semi-differentiability.

Example

Lemma 4 . 1 . 4 f 3 5.

 4143 Let h := 2 (u, x) g I(ū,x) (ū, x) Assume that for each M 2 @ x h(ū, x),w eh a v e M (R n )+{0}⇥R |I(ū,x)| = = R |I(ū,x)|+p .T h e nf o re a c hs e q u e n c e{(u n ,y n )}✓gphY with (u n ,y n ) ! (ū, ȳ),t h e r ee x i s t sx n ! x with x n 2 X(u n ) and y n = h(x n ,u n ) for sufficiently large n.M o r e o v e r , kx n -xkk(ku nūk + ky nȳk),

Theorem 4 . 5 .

 45 Let (ū, ȳ) 2 gphV and let Y be closed at ū.I ft h ef o l l o w i n gc o n d i t i o n s hold, then Y + R p = is pseudo-Lipschitz continuous at (ū, ȳ).

Theorem 4 . 6 .

 46 Let (ū, ȳ) 2 gphV .A s s u m et h a tt h ef o l l o w i n gc o n d i t i o n sh o l d :

Proposition 4 . 2 .

 42 Let (ū, ȳ) 2 gphV and let Y be closed at ū.A s s u m et h a tt h e r ei sa neighborhood N of ū such that Y (u) is convex for each u 2 N .I fY is pseudo-Lipschitz at (ū, ȳ),t h e nY 1 (ū) \-R p = = {0}.

Theorem 4 . 7 .

 47 Let Y be pseudo-Lipschitz at (ū, ȳ) 2 gphV .I ff o rs o m en e i g h b o r h o o d U ⇥ O of (ū, ȳ) the following conditions hold, then V is pseudo-Lipschitz at this point: (i) Y is closed on U and gphV is closed around (ū, ȳ).

(

  ii) DY (u, y)(0) \-R p = = {0} for all (u, y) 2 (U ⇥ O) \ gphV ; (iii) Y 1 (u) \-R p = = {0} for each u 2 U .Proof. We argue by contradiction. If it is false, then there exists (σ, 0) 2 N gphV (ū, ȳ)suchthat σ 6 =0.W ec o n s i d e rt w oc a s e s :e i t h e r(σ, 0) belongs to Fréshet normal cone or not. Let (σ, 0) 2 NgphV (ū, ȳ).B yd e fi n i t i o n , lim sup (u,y) gphV ---! (ū,ȳ) hσ, uūi kuūk + kyȳk  0. (4.6) Moreover, since Y is pseudo-Lipschitz at this point, (σ, 0) / 2 N gphV (ū, ȳ). This implies (σ, 0) / 2 NgphV (ū, ȳ). Thus, there are ✏>0 and {(u n ,y n )}✓gphY such that (u n ,y n ) ! (ū, ȳ) and lim n!1 hσ, u nūi ku nūk + ky nȳk > 2✏. (4.7)

lim n! 1 (

 1 u nū, y nr nȳ) kr n k =(0, -r),which is a contradiction with D Y (ū, ȳ)(0)\-R p = = {0}.H e n c e ,i f(✓, 0) 2 NgphV (ū, ȳ), then ✓ =0.Now, we consider the other case. By definition, there are sequences {(u n ,y n )}✓gphVand (σ n ,µ n ) ! (σ, 0) such that (u n ,y n ) ! (ū, ȳ) and (σ n ,µ n ) 2 NgphV (u n ,y n ).S i n c eY is pseudo-Lipschitz at (ū, ȳ),t h e r ei s✏>0 such that for each sufficiently large n, there exists a sequence {(u i n ,y i n )}✓gphY withlim i!1 hσ n ,u i nu n i + hµ n ,y i ny n i ku i nu n k + ky i ny n k

Proposition 4 . 4 .

 44 Let (ū, ȳ) 2 gphV .T h e nf o re a c hl 2 R q , D low V (ū, ȳ)(l) ✓{W min D low Y (ū, ȳ)(l)} (4.9)

Theorem 4 . 8 .

 48 Let (ū, ȳ) 2 gphV .A s s u m et h a tt h ef o l l o w i n gc o n d i t i o n sh o l d :(i) Y is locally dominated around ū and Y is closed at ū;(ii) DY (ū,ȳ) (0) \-R p = = {0}; (iii) Y 1 (ū) \-R p = = {0}.Then for each l 2 R q , min D low Y (ū, ȳ)(l) ✓ D low V (ū, ȳ)(l). (4.10) Proof. Let e 2{ min D low Y (ū, ȳ)(l)}.F r o m t h e a s s u m p t i o n s ,f o r t n # 0 and l n ! l,t h e r ei sas e q u e n c ee n ! e such

Example 4 . 6 .+ y 2 2 5 Definition 4 . 2 .Remark 4 . 3 .

 4654243 Let Y (u)={y 2 R 2 : y 2 1 1}. Y and V are strictly semidifferentiable and semi-differentiable at (ū, ȳ)=(0, (-1, 0)),r esp e ctively,andD low V (ū, ȳ)(1) = {e 2 R 2 : e 1 =0,e 2 5 0} withW min DY s-low Y (ū, ȳ)(1) = {e 2 R 2 = : e 1 =0},min DY s-low Y (ū, ȳ)(1) = ;. We cal l ȳ 2 V (ū) au n i f o r m l yp r o p e re ffi c i e n tp o i n t ,i ft h e r ei sa convex, closed and pointed coneK such that R n = ✓ int(K) [{0}.F u r t h e r m o r e ,f o r some neighborhood N of (ū, ȳ), Y (u) \ (y -K)={y}, 8(u, y) 2 gphV \ N.The next theorem gives some conditions under which the inclusion given in Theorem 4.8 holds by an equality. Proposition 4.5. If ȳ 2 V (ū) is a uniformly proper efficient point, then for each l 2 R q we have D low V (ū, ȳ)(l) ✓{min D low Y (ū, ȳ)(l)}. (4.11) Proof. Let l be given and e 2 D low V (ū, ȳ)(l).S u p p o s e t o t h e c o n t r a r y t h a t e/ 2 {min D low Y (ū, ȳ)(l)}. Then there is some e 0  e with e 0 2 D low Y (ū, ȳ)(l).L e t t n # 0 and d n ! d be given. By definition, there are e n ! e and e 0 n ! e 0 with {(ū + t n d n , ȳ + t n e n )}✓gphV and {(ū + t n d n , ȳ + t n e 0 n )}✓gphY . Thus, we have lim n!1 ȳ + t n e 0 n -(ȳ + t n e n )= lim n!1 e 0 ne n = e 0e. This contradicts the uniform proper efficiency hypothesis. The Propositions 4.4, 4.8 and 4.5 hold when D low is substituted by D, D low ,D s-low ,D s-up and D a providing Y is semi-differentiable at (ū, ȳ).

  (ū, ȳ) andD low V (ū, ȳ)(l)={min D low Y (ū, ȳ)(l)}.(4.12) Proof. The proof involves two parts. First, we show that e 2 DV (ū, ȳ)(l) implies e 2{ min D low Y (ū, ȳ)(l)}.S e c o n d ,w es h o wt h a te 2{ min D low Y (ū, ȳ)(l)} leads to e 2 D low V (ū, ȳ)(l). Let e 2 DV (ū, ȳ)(l).F r o mt h ed e fi n i t i o no ft h eB o u l i g a n dt a n g e n tc o n e ,t h e r ea r e sequence {(u n ,y n )}✓gphV such that {0} * (Y (u n )y n ) \-K ✏ ,

8 > 2 Y

 82 > > > > > > > < > > > > > > > > : {0} * (Y (u n )y n ) \-K ✏n , (u n ,y n ) ! (ū, ȳ),λ n ! λ, λ n = 0,λ j n =min 1ip λ i n  ✏ n , -λ n 2 N Y (un) (y n ), P p i=1 λ i n =1.Since Y is pseudo-Lipschitz at (ū, ȳ),f o ran e i g h b o r h o o dN of ȳ and for each z (ū) \ N ,t h e r ei sas e q u e n c e{(u n ,z n )}✓gphY with z n ! z. Thus, for sufficiently large n,hλ n ,z ny n i≥0. This implies -λ 2 N Y (ū)\N (ū).S i n c eY (u) is convex, -λ 2 N Y (ū) (ū) which contra-dicts normal efficiency of ȳ.

  + t n d n , ȳ + t n e n }✓Γ.

	Definition 1.17. As e t -v a l u e dm a p p i n gD low Γ(x, ȳ):R n ◆ R p is called lower Dini
	derivative of Γ at (x, ȳ) 2 gphΓ if for each d 2 R n ,
	D low Γ(x, ȳ)( d)=liminf d! d,t#0	Γ(x + td) -t	ȳ	.
	If T gphΓ (x, ȳ)=gphD low Γ(x, ȳ),t h e nΓ is said to be semi-differentiable at (x, ȳ);
	equivalently, if for each (d, e) 2 T gphΓ (x, ȳ), d n ! d and t n # 0,t h e r ee x i s t se n ! e
	such that {(x As mentioned above, the semi-differentiability is an extension of the lower Dini
	derivative. The concept of semi-differentiability was introduced by Penot [60]. He
	also established semi-differentiability of some parametric systems. It is obvious from
	definitions that semi-differentiability implies Proto-differentiability. The converse of
	this statement may not hold, in general. The following example demonstrates this.
	Example 1.4. Let Γ:R ◆ R be given by

  ! e such that {(x n + t n d n ,y n + t n e n )}✓Γ. Definition 1.[START_REF] Deb | Introducing robustness in multi-objective optimization[END_REF]. As e t -v a l u e dm a p p i n gD s Γ:R n ◆ R p is called s-derivative of Γ at (x, ȳ) 2 gphΓ,i fD s Γ( d) is consisted of all e such that there are (d n ,e n ) ! (d, e) and {t n }✓R ≥ with {(x + t n d n , ȳ + t n e n )}✓gphΓ and t n d n ! 0.

	As the last notion of the subsection, we address s-derivative. It was introduced
	by Shi under the name TP-derivative [67]. Taa called it s-derivative [69].
	addition, if T gphΓ (x, ȳ)=gphD s-low Γ(x, ȳ),thenwesayΓ is strictly semi-differentiable
	at (x, ȳ).I no t h e rw o r d s ,i ff o re a c h(d, e) 2 T gphΓ (x, ȳ), d n ! d, (x n ,y n )	gphΓ ---! (x, ȳ)
	and t n # 0,t h e nt h e r ee x i s t se n Rockafellar [2] established Proto-differentiability of a parametric system under
	Mangasarian-Fromovitz constraint qualification and Amahroq et al. [2] proved its
	strict semi-differentiability under the same condition. They also investigated strict
	semi-differentiability of general parametric systems involving intersection. It is obvi-
	ous that strict semi-differentiability at a given point implies both semi-differentiability
	and Clarke differentiability. however, the converse does not necessarily hold. The fol-
	lowing proposition introduces sufficient conditions under which strict semi-differentiability
	is derived from Clarke differentiability.
	Proposition 1.2. Let (x, ȳ) 2 gphΓ be given and let Γ be pseudo-Lipschitz con-
	tinuous at the point. If Γ is Clarke differentiable at (x, ȳ), then it is also strictly
	semi-differentiable at the point.

  2 ,...,λ m are nonnegative fixed scalars. This problem is an

	extension of Problem (2.2). Setting λ i =1 ,f o ri =1 , 2,...,p, Model (2.5) leads to
	Model (2.2).

The vector x 0 2 X is efficient if and only if x 0 2 X is an optimal solution of Problem (2.5); see

[START_REF] Ehrgott | Multicriteria optimization[END_REF][START_REF] Guddat | Multiobjective and Stochastic Optimization Based on Parametric Optimization,v o l .2 6 . A k a d e m i e -V e r l a g[END_REF]

. It is not difficult to see that the unboundedness of Problem

(2.5)

implies the unboundedness of Problem (2.2). Therefore, by Theorem 1, we get the following corollary about the problem of Guddat et al.

[START_REF] Guddat | Multiobjective and Stochastic Optimization Based on Parametric Optimization,v o l .2 6 . A k a d e m i e -V e r l a g[END_REF]

. Corrolary 2.1. If Problem (2.5) is unbounded, then X PE = ;.

  a fixed nonzero vector and x 0 is a given feasible solution. If x 0 is an efficient solution of Problem (2.1), then the optimal value of the preceding problem is zero. To the contrary, assume that there exists x 0 2 X such that Problem (2.6) is unbounded. Because of X PE 6 = ;,t h e r ei sx 2 X PE .A c c o r d i n gt oH e n i gp r o p e r efficiency, there exists a closed, convex and pointed cone K,suchthatC\{0}✓int(K)

	Theorem 2.3. If X PE 6 = ; with respect to C,t h e nf o ra n yx 0 2 X,P r o b l e m(2.6)
	has a finite optimal value.
	Proof.

  t) is an optimal solution of Problem (2.8),t h e nx is a weakly efficient solution of Problem (2.1).

(ii) If x is an efficient solution of Problem (2.1),t h e n(x, 0) is an optimal solution of the following problem for every r 2 R p = :

min t s.t. f (x) 5 f (x)+tr x 2 X.

Proof. See [26]. Proposition 2.2. If Problem (2.8) is unbounded for a 2 R p and r 2 R p ≥ ,t h e nm u l t iobjective optimization Problem (2.1) does not have any properly efficient solution.

  Theorem 3.12 gives a sufficient condition for robust efficient solutions of Problems (3.21) and (MOP) 0 to remain robust efficient for (MOP) ↵ . Theorem 3.12. If x is a robust efficient solution for both Problems (3.21) and (MOP) 0 ,t h e nx is a robust efficient solution of (MOP) ↵ for each ↵ 2 [0, 1]. Proof. Let ↵ 2 [0, 1]. By Theorem 3.10, x is efficient for Problem (MOP) ↵ .

	Now, we show that x is robust for (MOP) ↵ . By Theorem 3.4,
	pos	✓ p [
		i=1

  .5)Proof. The proof includes two parts. First we show that if (l, e) 2 T gphY +R p proof of Theorem 4.5, it can be shown that r n ! 0. Then, if t -1 n r n has a convergent subsequence, without loss of generality, we may assume that t -1 n r n ! r.H e n c e ,t -1 n y n ! ē 2 D s-low Y (ū,ȳ) (ζ), which implies e 2 D s-low Y (ū, ȳ)(ζ)+R p = ,for otherwise, t -1 n r n is unbounded. We may assume that kr n k -1 y n converges to a nonzero vector r 2-R p = . Therefore, limFor the second part, let e 2 D s-low Y (ū, ȳ)(l),d 2 R p = , l n ! l and {(u n ,y n + r n )}✓ gph(Y + R p = ) such that (u n ,y n + r n ) ! (ū, ȳ).Wec a ns h o ws i m i l a rt ot h ep r o o f of Theorem 4.5 that r n ! 0,a n dy n ! ȳ.S i n c e Y is strictly semi-differentiable at (ū, ȳ),t h e r ee x i s t se n ! e such that {(u n + t n l n ,y n + t n e n )}✓gphY .H e n c e , {(u n + t n l n ,y n + r n + t n e n )}✓gph(Y + R p

	n!1	(u n -ū,	y n kr n k	)=(0, r),
	which is impossible. Hence, t -1 n d n has a convergent subsequence and the first part is
	established.
				=	(ū, ȳ),
	then e 2 D s-low Y (ū, ȳ)(l)+R p = .N e x t ,w ee s t a b l i s ht h a te 2 D s-low Y (ū, ȳ)(l)+R p = implies e 2 D s-low (Y + R p = )(ū, ȳ)(l), which completes the proof.
	let (l, e) 2 T gph(Y +R p = ) (ū, ȳ).B y d e fi n i t i o n , t h e r e a r e s e q u e n c e s {(u n ,y n + r n )}✓ gph(Y + R p

= ) and t n # 0 such that t -1 n (u nū, y n + r nȳ) ! (l, e).S i m i l a rt ot h e = ),a n dt h ep r oo fi sc o m p l e t e .

  that {(ū + t n l n , ȳ + t n e n )}✓gphY .B y t h e assumptions, for sufficiently large n,w ecanconstructasequence{h n } such that{(ū+ t n l n , ȳ + t n h n )}✓gphV and ȳ + t n h n 5 ȳ + t n e n . The sequence {t n h n } tends to zero, since otherwise, either it tends to infinity or it has a convergent subsequence. Suppose that {t n h n } tends to infinity and kh n k -1 h n ! h.S i n c eh n 5 e n ,wehaveh  0. Thus, Similar to the former case it is shown that r 2 R p ≥ .H e n c e , ȳr 2 Y (ū) which contradicts ȳ 2 V (ū). Thus, {t n h n } tends to zero. The proof is complete by showing that {h n } tends to e. Assume that this is not the case. Then, either it tends to infinity or it has a subsequence not converging to e.I f{h n } tends to infinity, then we may assume that kh n k -1 h n ! h 2-R p ≥ . Consequently, which contradicts condition (ii). Now, without loss of generality, assume that h n ! h with h n 6 = e.B yv i r t u eo ft h ec o n s t r u c t i o no f{h n }, h  e. Thus,

	lim n!1	(	ū + t n l n -t n kh n k	ū	,	ȳ + t n h n -t n kh n k	ȳ	)=(0,h),
	lim n!1	(	ū + t n l n -t n	ū	,	ȳ + t n h n -t n	ȳ
		lim n!1	(ū + t n l n ,	ȳ + t n h n t n kh n k	)=(0,h),

which contradicts condition (iii). For the second case, without loss of generality, we assume that t n h n !-r.

Acknowledgments

I would like to express my deepest gratitude to my supervisors, Prof. Dinh The Luc and Prof. Majid Soleimani-damaneh, for their guidance and continuous support throughout my doctorate study. Their patience and helpful assistance helped me finish this thesis. Finally, I would like to thank my parents for understanding, love and support.

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where ✏ 2 R p .T h e nx is a set-based minmax robust weakly efficient solution. In addition, if x is the unique optimal solution, then it is a set-based minmax robust efficient solution.

Unlike deterministic multi-objective optimization, the aforementioned methods cannot produce all set-based minmax robust (weakly) efficient solutions even for the convex case; see [START_REF] Ehrgott | Minmax robustness for multiobjective optimization problems[END_REF] for some counterexamples.

In general there is no relationship between set-based minmax robust efficiency and flimsily robust efficiency. Strictly speaking, a feasible point may be set-based minmax robust efficient while it is not flimsily robust efficient, and vice verse. Similarly highly robust efficiency and set-based minmax robust efficiency have no connection. To clarify this point, we take another look at Example 3.1. x 3 is a set-based minmax robust efficient point while it is neither flimsily robust efficient nor highly robust efficient.

Based on the idea of set-based minmax robust efficiency, the concept of hull-based minmax robust efficiency has been proposed by Bokrantz et al. [START_REF] Bokrantz | On solutions to robust multiobjective optimization problems that are optimal under convex scalarization[END_REF].

Definition 3.4. The decision variable x 2 X is called hull-based minmax robust efficient if there is no x 2 X such that

It is established that hull-based minmax robust efficiency implies set-based minmax robust efficiency [START_REF] Bokrantz | On solutions to robust multiobjective optimization problems that are optimal under convex scalarization[END_REF]. However, the converse does not hold necessarily; for instance, in Example 3.1, x 3 is set-based minmax robust efficient but not hull-based minmax robust efficient. Another notion suggested for robustness in multi-objective optimization is point-based minmax robust efficiency, which deals with efficient solutions of a deterministic multi-objective optimization problem for which each objective function is worst case of the original objective on the uncertain set.

Consider the following multi-objective optimization problem:

where X ✓ R n is nonempty and f : X ! R p is locally Lipschitz. Note that the notation k.k stands for the Frobenius norm in matrix spaces. However, since all the norms are equivalent in finite dimension, the definition of robust is independent of the choice of norm. In the above definition, ✏ is called radius of robustness. Thanks to mean value Theorem 1.5, we have the following result.

Proposition 3.3. Let x be a robust efficient solution with radius ✏.T h e nf o ra n y Lipschitz function h : R n ! R p with modulus less than ✏, x is an efficient solution of the following problem:

The following theorem presents a nice property of robust efficient solutions. It states that the set of robust efficient solutions is a subset of properly efficient solutions under the compactness of the feasible set. Proof. Suppose x is not a properly efficient solution. Then, there exist {x i }✓X, increasing sequence {M i } of positive real numbers, and k 2{ 1,...,p},s u c ht h a t

Robustness radius

In this short section, we compute a radius of robustness. For a given vector a 2 R p , the vector a + is obtained from a by substituting all negative components by zero. It is not difficult to show that ka + k is equal to the distance from a to -R p = = {x 2 R p : x  0}. Lemma 3.1. Let X be a closed and convex set and f i (i =1,...,p)b ec onvex. L etd 2

and it is equal to the optimal value of the following problem,

Proof. First, we show that f 0 (x; d) / 2-R p = .I f f 0 (x; d)  0,t h e nd u et ot h e convexity of f ,w eh a v ed 2 G(x), which gives a contradiction according to Theorem 3.2. The proof of the second part is similar to that of Lemma 4.2 of [START_REF] Georgiev | Robust aspects of solutions in deterministic multiple objective linear programming[END_REF]. Theorem 3.7. Under the assumptions of Lemma 3.1, the optimal value of the following problem is positive and it is a robustness radius for x:

Proof. Let ⇢ be the optimal value of the given problem. Thus, by Lemma 3.1, ⇢>0. Now, we show that ⇢ is a robustness radius for x.I fi ti sn o tar o b u s t n e s s radius, then there exist some x o 2 X and some matrix

with uncertain set U = {C p⇥n : kCk <✏} and uncertain objective f (C, x)=f (x)+Cx.

Proof. The proof is not difficult and is hence omitted.

The following proposition provides a connection between the robustness and pointbased minmax robust efficiency.

These relations contradict the robustness of x (in the sense of Definition 3.6) and the proof is complete.

The next theorem states relationship between robust efficiency and set-based minmax robust efficiency. Moreover, we also investigate the connection with hull-based exist λ 2 R p and w 2 R m such that

Therefore, by assumption of the theorem, 0 2 P p i=1 λi @f i (x)+ P m j=1 wi @g j (x), for some λ>0, w ≥ 0. This implies that x is a properly efficient solution of Problem (3.21).

By a manner similar to the proof Theorem 3.13, it can be shown that the result is valid for weak efficient solutions as well. The following example shows that this result may not be valid for efficient solutions.

Example 3.4. Let g(x)=f (x)=x and h(x)=x 2 .I ti sc l e a rt h a tx =0is an efficient solution of (MOPh) and {rh(0)}✓pos(rf (0)) but x =0is not an efficient solution of Problem (3.21).

The following result gives even more insights into the connection between the proper efficient solutions of the two problems (3.21) and (MOPh), when the f i and g j functions are continuously differentiable. (MOPh),t h e nt h e r ee x i s tv e c t o r su 2 R p and v 2 R m such that u>0,a n d

(ii) Let x be a properly efficient solution of Problem (3.21). If there exist vectors

then x is a properly efficient solution of Problem (MOPh).

Proof. (i) Since x is a properly efficient solution of Problem (MOPh),thenthere efficient set and so on; see, e.g., [START_REF] Luc | Theory of vector optimization[END_REF][START_REF] Luc | Multiobjective Linear Programming[END_REF][START_REF] Sawaragi | Theory of multiobjective optimization[END_REF][START_REF] Tanino | Sensitivity analysis in mcdm[END_REF] and the references therein. Moreover, Sawaragi et al. [START_REF] Sawaragi | Theory of multiobjective optimization[END_REF] also investigate the problems with parameter-dependent ordering cones. Bednarczuk [START_REF] Bednarczuk | Stability Analysis for Parametric Vector Optimization Problems[END_REF][START_REF] Bednarczuk | On lower lipschitz continuity of minimal points[END_REF] introduces the concept of strict efficiency and she establishes Holder like continuity of feasible solution set and optimal value under strict efficiency and some other appropriate assumptions.

Some scholars study stability from a different standpoint. They consider a sequence of vector optimization problems whose feasible set and objective function tend to a given set and a given function in the sense of Painlevé-Kuratowski or other sense, respectively. They give some sufficient conditions for convergence of the sequence of efficient sets to the efficient set of the given problem; see [START_REF] Li | Stability results for properly quasi convex vector optimization problems[END_REF][START_REF] Miglierina | Convergence of minimal sets in convex vector optimization[END_REF] and the references therein.

Sensitivity analysis is also a strong tool in post-optimal analysis. There are numerous works devoted to this concept in vector optimization. Tanino [START_REF] Tanino | Sensitivity analysis in multiobjective optimization[END_REF][START_REF] Tanino | Stability and sensitivity analysis in convex vector optimization[END_REF] is among the first scholars who have dedicated some studies to this issue. He applies contingent derivative for analyzing efficient solutions and efficient values [START_REF] Tanino | Sensitivity analysis in multiobjective optimization[END_REF][START_REF] Tanino | Stability and sensitivity analysis in convex vector optimization[END_REF]. Shi [START_REF] Shi | Contingent derivative of the perturbation map in multiobjective optimization[END_REF] introduces the concept of TP cone. Some results on contingent derivative of the efficient value mapping are improved and extended by using this concept [START_REF] Kuk | Sensitivity analysis in vector optimization[END_REF][START_REF] Shi | Contingent derivative of the perturbation map in multiobjective optimization[END_REF]. We refer the readers to excellent survey [START_REF] Tanino | Sensitivity analysis in mcdm[END_REF].

Lee et al. [START_REF] Lee | On sensitivity analysis in vector optimization[END_REF] use Proto derivative for analyzing parametric vector optimization.

The authors obtain some sufficient conditions for Proto differentiability of efficient solution set and efficient values. Chuong et al. [START_REF] Chuong | Generalized clarke epiderivatives of parametric vector optimization problems[END_REF] use the generalized Clarke epiderivative notion to study the problem. They provide some formulas for calculating this derivative. Chuong [START_REF] Chuong | Derivatives of the efficient point multifunction in parametric vector optimization problems[END_REF] studies the sensitivity analysis via s-derivative. He also gives some formulas for inner and outer estimation of s-derivative for efficient values.

The concept of coderivative has also been applied to investigating the behavior of efficient solutions and efficient values; see [START_REF] Chuong | Clarke coderivatives of efficient point multifunctions in parametric vector optimization[END_REF][START_REF] Khan | Set-valued Optimization[END_REF] and the references therein.

Here, we study parametric multi-objective optimization problem whose feasible set is given explicitly by some inequalities. Throughout this chapter, all functions are assumed to be locally Lipschitz. We establish some theorems for strict semidifferentiability of feasible set and feasible values. Moreover, we give some formulas Thus, X is not strictly semi-differentiable at (ū, x).H e r e ,g is not regular.

Example 4.2. Let g(u, x)=x 2u 2 .I tc a nb ev e r i fi e de a s i l ya t(ū, x)=(0, 0) that

Thus, X is not strictly semi-differentiable at (ū, x).H e r e ,C Qd o e sn o th o l da tt h e given point.

In the rest of the section, we deal with pseudo-Lipschitz continuity and strict semidifferentiability of Y at a given point. To express the result more concisely, we define the set-valued mapping X : gphY ◆ R n given by X(u, y)={x 2 X(u):f (u, x)=y}. (i) there is x 2 X(ū, ȳ) such that for any sequence {(u n ,y n )}✓gphY with (u n ,y n ) ! (ū, ȳ),t h e r ee x i s t sas e q u e n c e{x n } with x n 2 X(u n ,y n ) admitting x as a cluster point and X is pseudo-Lipschitz continuous at (ū, x);

(ii) the set-valued mapping X is locally compact on N \gphY for some neighborhood N of (ū, ȳ). X is pseudo-Lipschitz continuous at each (ū, x) 2 X(ū, ȳ).

Proof. According to Theorem 1.

with positive sequence {✏ n } tending to zero. We can infer from Theorem 1.2 that (σ, 0) 2 N gphX (ū, x). This is a contradiction with pseudo-Lipschitz continuity of X at (ū, x),a n ds oY is pseudo-Lipschitz continuous at (ū, ȳ).

The pseudo-Lipschitz continuity of Y at (ū, ȳ) under (ii) can be proved in a similar way.

It is easy to see that if [ u2U X(u) is bounded for some neighborhood U of ū,t h e n the first part of condition (ii) in Theorem 4.1 is fulfilled. In the following theorem, we provide another hypothesis for having Theorem 4.1. Before we state the theorem, we need to recall some notions.

Consider the set-valued mapping X. The outer horizon limit of X at ū is denoted and defined by X 1 (u): ={lim n t n x n : t n # 0,u n ! u, x n 2 X(u n )}. The outer horizon limit of a set-valued mapping at a given point is a closed cone. The reader can see [START_REF] Rockafellar | Variational analysis,v o l .3 1 7[END_REF] for more information about the outer horizon limit.

The asymptotic function of f with respect to the set-valued mapping X at u is defined and denoted by

with some modifications). In general, asymptotic function is a closed-valued setvalued mapping. We refer the reader to [START_REF] Luc | Convergence of asymptotic directions[END_REF] for more details. Theorem 4.2. Let (ū, ȳ) 2 gphY be given and let gphY be closed around (ū, ȳ).

Assume the following conditions hold:

(i) X is pseudo-Lipschitz continuous at (ū, x) for every x 2 X(ū, ȳ);

Then Y is pseudo-Lipschitz continuous at (ū, ȳ).

Proof. For each sequence {(u n ,y n )}✓gphY with (u n ,y n ) ! (ū, ȳ),t h e r ee x i s t s as e q u e n c e{x n } with x n 2 X(u n ,y n ).I f {x n } has no cluster point, then {kx n k} converges to 1.B yt a k i n gt n =1 /kx n k,w em a ya s s u m e{t n x n } converges to some nonzero vector d. Then

0= lim

n!1

gphY and t n # 0 with t -1 n (u nū, y nȳ) ! (l, e). According to the assumptions (without relabelling), there is a sequence x n 2 X(u n ) with y n = f (u n ,x n ) and x n ! x.

Consider the sequence of (x nx)/t n .I f i t i s n o t b o u n d e d ,t h e n w e m a y a s s u m e (x nx)/kx nxk converges to some nonzero vector d and t n /kx nxk converges to zero as n tends to 1. Then we get

which contradicts the hypotheses. Hence, (x nx)/t n is bounded and without loss of generality we assume that t -1 n (x nx) ! d.A sX is strictly semi-differentiable at (ū, x), d 2 D s-low X(ū, x)(l). Moreover, due to the regularity of f at this point e = f 0 (ū, x; l, d). Now, we show that if e = f 0 (ū, x; l, d) for some d 2 D s-low X(ū, x)(l),t h e ne 2 D s-low Y (ū, ȳ)(l). Consider an arbitrary sequences {(u n ,y n )}✓gphY and {t n } with (u n ,y n ) ! (ū, ȳ) and t n # 0.B y ( i ) , t h e r e i s a s u b s e q u e n c e {x n k } with x n k 2 X(u n k ,y n k ) and x n k ! x. From strictly semi-differentiability of X at (ū, x),thereare

}✓gphY ,a n dr e g u l a r i t y of f at (ū, x) implies e n k ! e.I nv i e wo fL e m m a1 . 1 ,w eh a v e(l, e) 2 T c gphY (ū, ȳ). From Theorem 4.1 and Proposition 1.2, the set-valued mapping Y is strictly semidifferentiable at the given point. This establishes the result.

In the next theorem, we provide another sufficient condition for the strict semidifferentiability of Y at a given point.

(ii) X is pseudo-Lipschitz continuous and strictly semi-differentiable on X(ū, ȳ) and f is regular on this set;

Proof. Suppose the assertion of the proposition is false. If we have a nonzero vector

B yd e fi n i t i o n ,t h e r ee x i s ts e q u e n c e s{(u n ,y n )}✓ gphY and t n # 0 with u n ! ū and t n y n ! e. This implies (t n (u nū),t n (y nȳ)) ! (0,e) which contradicts our assumption. Thus, the proof is complete.

Shi [START_REF] Shi | Contingent derivative of the perturbation map in multiobjective optimization[END_REF] gave some conditions under which D s Y (ū, ȳ)(0) \-R p = = {0}; see Proposition 3.2. We have this property when gph(Y + R p = ) is convex; see [START_REF] Tanino | Sensitivity analysis in mcdm[END_REF]. As a result, the mentioned conditions imply

The following example shows that the hypotheses (ii) and (iii) in Theorems 4.5 and 4.6 are essential for having pseudo-Lipschitz continuity and strict semi-differentiability, respectively.

Example 4.5. Let

Note that Y is pseudo-Lipschitz and strictly semi-differentiable at (0, 0),b u tY + R = is neither pseudo-Lipschitz nor strictly semi-differentiable at this point. Indeed,

We refer the reader to Example 3.4 of [START_REF] Tanino | Sensitivity analysis in multiobjective optimization[END_REF] to see that proper efficiency and closedness are also necessary in Theorem 4.6.

In general, pseudo-Lipschitz continuity and strict semi-differentiability of Y and Y + R p = are not related to each other. Y may be pseudo-Lipschitz continuous and strictly semi-differentiable at a given point, while Y + R p = does not fulfill these properties at the given point, and vice versa. To check pseudo-Lipschitz and strict semi- The following lemma gives some conditions to guarantee local domination property for Y near a given point.

where n 0 2 N.S i n c eh  e,f o rs u ffi c i e n t l yl a r g en, this leads to a contradiction with (4.13).

The rest of proof follows from that of Theorem 4.8.

In Theorem 4.9 if one substitutes semi-differentiability of Y for prorto-differentiability, then prorto-differentiability of V follows in a similar manner. This is stronger than the conditions given by Lee-Huy [START_REF] Lee | On sensitivity analysis in vector optimization[END_REF], only in the convex case. Tanino [START_REF] Tanino | Stability and sensitivity analysis in convex vector optimization[END_REF] proposed the notion of normal efficiency for the case that Y is a convex set-valued mapping (gphY is a convex set). He proved the relation (4.12) with respect to contingent derivative under the convexity condition and normal efficiency. Theorem 4.9 can be adapted such that we have Tanino's result without convexity and normal efficiency.

In the following lemma, we extend the definition of normal efficiency for the case that Y is convex-valued. Furthermore, we show that normal efficiency implies uniformly proper efficiency even for this extension. Proposition 4.6. Let (ū, x) 2 gphE and let CQ1 hold at this point. If V is pseudo-Lipschitz continuous at (ū, f (ū, x)),t h e nE is pseudo-Lipschitz at the given point.

Proof. Since CQ1 holds at (ū, x), X is pseudo-Lipschitz at this point. Thus, pseudo-Lipschitz continuity of E at (ū, x) follows from pseudo-Lipschitz continuity of V . (i) f is regular at (ū, x);

(ii) CQ1 holds at (ū, x);

(iii) V is pseudo-Lipschitz and semi-differentiable at (ū, f (ū, x)) and gphV is closed around it.

Then, E is semi-differentiable at (ū, x) and D low E(ū, x)(l)={d 2 D s-low X(ū, x)(l):C 0 (ū, x; l, d) 2 D low V (ū,ȳ) (l)} (4.14)

Proof. E(u) can be shown as {x 2 R n : g(u, x) 2 K}, where g(u, x)= differentiability can be substituted by regularity). Thus, E is Proto differentiable at (ū, x) and we have (4.14). Since E is pseudo-Lipschitz at the given point, it is also semi-differentiable at (ū, x).