Nguyen Thi

Hong Linh
email: honglinh98t1@yahoo.com

Jean-Paul Penot
email: jean-paul.penot@univ-pau.fr

Generalized convex functions and generalized di¤erentials

Keywords: colinvex, generalized di¤erential, mathematical programming, optimality conditions, protoconvex function, pseudoconvex function, quasiconvex function. Mathematics Subject Classi…cation: 26B25, 46G05, 49K27, 90C26, 90C32

We study some classes of generalized convex functions, using a generalized di¤erential approach. By this we mean a set-valued mapping which stands either for a derivative, a subdi¤erential or a pseudo-di¤erential in the sense of Jeyakumar and Luc. We establish some links between the corresponding classes of pseudoconvex, quasiconvex and another class of generalized convex functions we introduced. We devise some optimality conditions for constrained optimization problems. In particular, we get Lagrange-Kuhn-Tucker multipliers for mathematical programming problems.

Introduction

Various needs have led mathematicians to introduce and study several notions of generalized convexity or concavity. Among the tools used to de…ne or study these notions are the various subdi¤erentials of nonsmooth analysis ( [START_REF] Aussel | Subdi¤ erential properties of quasiconvex and pseudoconvex functions: Uni…ed approach[END_REF], [START_REF] Hadjisavvas | Handbook of generalized convexity and generalized monotonicity[END_REF], [START_REF] Hassouni | On generalized monotone multifunctions with applications to optimality condition in generalized convex programming[END_REF], [START_REF] Luc | Handbook of generalized convexity and generalized monotonicity[END_REF], [START_REF] Penot | Are generalized derivatives useful for generalized convex function[END_REF], [START_REF] Penot | Characterization of solution sets of quasiconvex programs[END_REF], [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF], [START_REF] Penot | Generalized convexity of functions and generalized monotonicity of set-valued maps[END_REF], [START_REF] Penot | Generalized monotonicity of subdi¤ erentials and generalized convexity[END_REF]...), the convexi…cators of [START_REF] Demyanov | Convexi…cation and concavi…cation of a positive homogeneous function by the same family of linear functions[END_REF], the pseudo-di¤erentials of Jeyakumar and Luc ( [START_REF] Jeyakuma | Nonsmooth vector functions and continuous optimization[END_REF]), the normal cones to sublevel sets ( [START_REF] Aussel | Normal cones to sublevel sets: an axiomatic approach. Applications in quasiconvexity and pseudoconvexity[END_REF], [START_REF] Aussel | Adjusted sublevel sets, normal operator, and quasi-convex programming[END_REF], [START_REF] Aussel | Quasiconvex programming with locally starshaped contraints region and applications to quasiconvex MPEC, Optimization[END_REF], [START_REF] Borde | Continuity properties of the normal cone to the level sets of a quasiconvex function[END_REF]) and the generalized directional derivatives ( [START_REF] Komlósi | Generalized monotonicity and generalized convexity[END_REF], [START_REF] Komlósi | Handbook of generalized convexity and generalized monotonicity[END_REF], [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF], [START_REF] Sach | Characterizations of generalized convexities via generalized directional derivatives[END_REF]). In the present paper we use a concept of generalized derivative which can encompass all these notions but the last one. We call it a generalized di¤erential in contrast with the notion of subdi¤erential because it is not necessarily a one-sided concept. It allows much ‡exibility. It also leads us to get rid of some assumptions required in previous works such as smooth renorming of the space. We establish comparisons with the notions obtained using directional derivatives or their substitutes.

The case of pseudo-a¢ ne functions (also called pseudo-linear functions) is studied elsewhere ( [START_REF] Linh | Generalized a¢ ne functions and generalized di¤ erentials[END_REF]) with similar concepts and methods. It requires some results of the present paper.

In the two last sections of the paper, we investigate some necessary and su¢ cient optimality conditions for constrained problems and mathematical programming problems.

Characterizations of generalized convex functions

We devote this preliminary section to review some concepts of generalized convexity; we also introduce new ones. Some elementary properties and characterizations are given.

In the sequel X, Y are normed vector spaces, X is the dual space of X, C is a nonempty subset of X and f : C ! R 1 := R [ f+1g: We extend f to XnC by +1; alternatively, we may consider f : X ! R 1 and take for C a subset of X containing the domain of f: We denote by P (resp. R + ) the set of positive numbers (resp. non negative numbers) and R stands for R [ f 1; +1g: We assume that a set-valued map @f : C X is given which stands as a substitute to the derivative of f ; we call it a generalized di¤ erential of f . Among possible choices for @f are the subdi¤erentials of f in the various senses of nonsmooth analysis and the pseudo-di¤erentials of f in the sense of [START_REF] Jeyakuma | Nonsmooth vector functions and continuous optimization[END_REF]. These cases will be considered below. We also deal with some other cases, such as normal cones set-valued maps.

We recall that the visibility cone V (C; x) of C at x 2 C is the cone generated by C x:

V (C; x) := P(C x) := fr(c x) : r 2 P; c 2 Cg:

It contains the radial tangent cone to C at x which is the set

T r (C; x) := fu 2 X : 9 (r n ) ! 0 + ; x + r n u 2 C 8ng:
The visibility bundle (resp. the radial tangent bundle) of C is the set

V C := f(x; u) 2 C X : 9r 2 P; w 2 C; u = r(w x)g = [ x2C fxg V (C; x) (resp. T r C := f(x; u) 2 C X : 9 (r n ) ! 0 + ; x + r n u 2 C 8ng = [ x2C fxg T r (C; x) ).
If C is starshaped at x (in the sense that for all w 2 C and t 2 [0; 1] one has

(1 t)w + tx 2 C) one has V (C; x) = T r (C; x): Thus, if C is convex, V C = T r C:
The upper and the lower radial derivatives (or upper and lower Dini derivatives) of f at x 2 C in the direction u 2 T r (C; x), are de…ned by

D + f (x; u) = lim sup t!0+; x+tu2C 1 t [f (x + tu) f (x)]; D + f (x; u) = lim inf t!0+; x+tu2C 1 t [f (x + tu) f (x)]:
The upper and the lower radial subdi¤erentials are de…ned by @ r f := fx : hx ; ui D + f (x; u) 8u 2 T r (C; x)g; @ r f := fx : hx ; ui D + f (x; u) 8u 2 T r (C; x)g:

The de…nitions of @f -pseudoconvexity and @f -quasiconvexity we adopt here for a generalized subdi¤erential @f of f are similar to the ones used for a subdi¤erential by several authors; see [START_REF] Penot | Are generalized derivatives useful for generalized convex function[END_REF], [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF] and the references therein. We also introduce de…nitions of @f -protoconvexity and strict @f -pseudoconvexity of f as natural variants of the two preceding concepts.

De…nition 1 Let f : C ! R 1 , x 2domf . A function f is said to be (i) @f -pseudoconvex at x if for all w 2 C : f (w) < f (x) ) for all x 2 @f (x) : hx ; w xi < 0: (1)
(ii) @f -quasiconvex at x if for all w 2 C :

f (w) < f (x) ) for all x 2 @f (x) : hx ; w xi 0: (2)
(iii) @f -protoconvex at x if for all w 2 C :

f (w) f (x) ) for all x 2 @f (x) : hx ; w xi 0: (3)
(iv) strictly @f -pseudoconvex at x if for all w 2 Cnfxg :

f (w) f (x) ) for all x 2 @f (x) : hx ; w xi < 0: (4)
We add the word "eventually" when in the preceding implications "for all" is changed into "there exists". Thus, for instance, f is eventually @f -pseudoconvex at x if for all w 2 C satisfying f (w) < f (x) there exists x 2 @f (x) such that hx ; w xi < 0:

Remark that the de…nition of @f -strict pseudoconvexity is irrealistic if x is a non unique minimizer of f: Clearly

f is strictly @f -pseudoconvex at x =) f is @f -protoconvex and @f -pseudoconvex at x; (5) f is @f -pseudoconvex at x =) f is @f -quasiconvex at x; (6) f is @f -protoconvex at x =) f is @f -quasiconvex at x: (7)
We say that f is @f -pseudoconvex on C (in short @f -pseudoconvex) if it is @f -pseudoconvex at every x 2 C: We use a similar convention for the other concepts introduced above.

Let us observe that @f -quasiconvexity is di¤erent from full @f -quasiconvexity (in the terminology of [START_REF] Penot | Are generalized derivatives useful for generalized convex function[END_REF]) in the sense (due to D. Aussel [START_REF] Aussel | Subdi¤ erential properties of quasiconvex and pseudoconvex functions: Uni…ed approach[END_REF]) that domf is convex and for any w; x 2 C; x 2 @f (x) with hx ; w xi > 0; one has f (w) f (z) for any z 2 [x; w]. Clearly, if f is @f -quasiconvex and quasiconvex then f is fully @f -quasiconvex while a fully @f -quasiconvex function is obviously @f -quasiconvex.

The following examples show that versatility is gained by taking for @f a multimap which may di¤er from usual subdi¤erentials such as the Fenchel, the Fréchet or the Dini-Hadamard subdi¤erentials. Example 1. Let f : R ! R be given by f

(x) = 2x for x 2 [ 1; 1]; f (x) = x + 1 for x > 1; f (x) = x 1 for x < 1.
We can take @f (x) = @ D f (x) when x 6 = 1 and @f (1) = @ D ( f )(1) (where @ D f is the Dini-Hadamard subdi¤erential of f ). Observe that @ D f (1) = ?. Example 2. Let f : X ! R be such that for some generalized di¤erential @ ] one has @ ] ( f )(x) 6 = ? for all x 2 X: Then it may be convenient to take @f := @ ] ( f )(x): Example 3. Let f : X ! R be such that f = f 1 + f 2 : Then one may take @f := @ ] f 1 + @ ] f 2 ; where @ ] is some generalized di¤erential, althougth it may be di¤erent from @ ] (f 1 + f 2 ): Example 4. Let f : X ! R be such that f = g h for some functions g; h: Then one may take @f (x) := @ ] g(x) @ ] h(x); where @ ] is some generalized di¤erential, and for subsets A; B of X , A B := fx 2 X : B + x Ag; althougth @f may be di¤erent from @ ] (g h): For some subdi¤erentials the preceding de…nitions may be automatically satis…ed. This is the case if @f is the Fenchel-Moreau subdi¤erential @ F M f given by @ F M f (x) := fx 2 X : f x + f (x) x (x)g. Let us consider some other classical subdi¤erentials. We recall that the original Greenberg-Pierskalla's subdi¤ erential is de…ned by

@ GP f (x) = fx 2 X : hx ; w xi 0 ) f (w) f (x)g:
The lower subdi¤ erential, or Plastria subdi¤ erential of f at some point x of its domain domf := fx 2 X : f (x) 2 Rg is the set

@ < f (x) := fx 2 X : 8w 2 S < f (x); f (w) f (x) hx ; w xig where S < f (x) := S < f (f (x)) := f 1 (( 1; f (x))
) is the strict sublevel set of f at x: We also recall the following variant, called the infradi¤ erential or Gutiérrez subdi¤ erential :

@ f (x) := fx 2 X : 8w 2 S f (x); f (w) f (x) hx ; w xig; where S f (x) := S f (f (x)) := f 1 (( 1; f (x)])
is the sublevel set of f at x: For such subdi¤erentials one has the following obvious statement.

Lemma 2 Let f : X ! R 1 and x 2domf: Then (a) f is @ GP f -pseudoconvex at x; (b) f is @ < f -pseudoconvex at x; (c) f is @ f -protoconvex and @ f -pseudoconvex at x.
We denote by N (C; x) the normal cone at x 2 X to a subset C of X given by N (C; x) := fx 2 X : hx ; w xi 0 8w 2 Cg even when C is nonconvex; of course, such a cone is mainly of interest in the case C is convex. Some normal cone operators can be associated to a function f : X ! R 1 as follows: for x 2domf;

N f (x) := N (S f (x); x) := fx 2 X : hx ; w xi 0 8w 2 S f (x)g; N < f (x) := N (S < f (x); x) := fx 2 X : hx ; w xi 0 8w 2 S < f (x)g:
(see for example [START_REF] Aussel | Adjusted sublevel sets, normal operator, and quasi-convex programming[END_REF], [START_REF] Borde | Continuity properties of the normal cone to the level sets of a quasiconvex function[END_REF], [START_REF] Penot | Are generalized derivatives useful for generalized convex function[END_REF] for the case f is quasiconvex). Clearly, any function f is N f -protoconvex and N < fquasiconvex at any x 2domf: Moreover, for any function f; @ GP f (resp.

N f , N < f ) is the greatest generalized di¤erential @f of f; such that f is @f -pseudoconvex (resp. @f -protoconvex, @f -quasiconvex).
Recall that f is said to be semi-strictly quasiconvex if it is quasiconvex and for all w; x; y 2 C, x 2 (w; y) one has f (x) < f (y) whenever f (w) < f (y):

Lemma 3 Let f : X ! R 1 be a quasiconvex function and let x 2 C such that f (x) > inf f (X). If there is no local minimizer of f on L f (x) := f 1 (f (x)), in particular f is semi-strictly quasiconvex, then N < f (x) = N f (x). Moreover, if f is radially upper semicontinuous, then f is N f nf0g-pseudoconvex at x:
This last assertion is given in [2, Proposition 3] and [5, Proposition 2.7]; we give a direct, simple proof for the convenience of the reader.

Proof. Since there is no local minimizer of f on L f (x) (that occurs when f is semi-strictly quasiconvex),

S < f (x) is dense in S f (x); hence N < f (x) = N f (x).
Then, for all x 2 N f (x)nf0g and w 2 S < f (x) we have hx ; w xi < 0 since if hx ; w xi = 0; taking u 2 X with hx ; ui > 0 we would have w + tu 2 S < f (x) for t > 0 small enough, and hx ; w + tu xi > 0; a contradiction. Thus f is N f nf0g-pseudoconvex at x:

We say that @f is a valuable generalized di¤ erential if for all a 2 X and b 2

X with f (b) > f (a) there exist c 2 [a; b) := [a; b]nfbg and sequences (c n ) ; (c n ) such that (c n ) ! f c; c n 2 @f (c n ) for each n and hc n ; d c n i > 0 for all d 2 b + R + (b a) and all n 2 N.
The following result shows that for a valuable subdi¤erential, quasiconvexity, @f -quasiconvexity and @fprotoconvexity coincide under a mild continuity assumption. The last assertion follows from [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF].

Proposition 4 (a) If f : X ! R 1 is quasiconvex and @f @ r f; then f is @f -protoconvex hence @f - quasiconvex. (b) Let f : X ! R 1 be a radially continuous l.s.c. function and let @f be a valuable generalized di¤ erential. If f is @f -quasiconvex, then it is quasiconvex. Proof. (a) Let w; x 2 C and x 2 @f (x) be such that hx ; w xi > 0. Then there exists (t n ) ! 0 + such that f (x + t n (w x)) > f (x): Since f is quasiconvex, one has f (w) > f (x):
(b) It follows from [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF] with a similar proof.

The following proposition is well known when @f is the derivative of f or a subdi¤erential of f .

Proposition 5

Let C be open, let f : C ! R be radially upper semicontinuous and 0

= 2 @f (x). (a) If f is @f -quasiconvex at x then it is @f -pseudoconvex at x. (b) If f is eventually @f -quasiconvex at x then it is eventually @f -pseudoconvex at x. Proof. Let w; x 2 C be such that f (w) < f (x): (a) If f is @f -quasiconvex at x
then for all x 2 @f (x); hx ; w xi 0: Suppose that there exists x 2 @f (x) such that hx ; w xi = 0: Since x 6 = 0; we can …nd u 2 X such that hx ; ui = 1: By radial upper semicontinuity of f; there exists " > 0 such that y := w + "u 2 C and f (y) < f (x): Since f is @f -quasiconvex, 0 hx ; y xi = hx ; w xi + "hx ; ui = " > 0; a contradiction. Hence, one has hx ; w xi < 0 for all x 2 @f (x): f is @f -pseudoconvex at x.

(b) If f is eventually @f -quasiconvex at x, there exists x 2 @f (x); such that hx ; w xi 0: With a similar proof to the one in (a), one gets hx ; w xi < 0.

A comparison between @f -pseudoconvexity, quasiconvexity (in the usual sense) and semi-strict quasiconvexity is given in the next proposition. Similar comparisons have been made in [START_REF] Hadjisavvas | Handbook of generalized convexity and generalized monotonicity[END_REF]Prop. 11.5], [START_REF] Penot | What is quasiconvex analysis?[END_REF], [START_REF] Sach | Characterizations of generalized convexities via generalized directional derivatives[END_REF], [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF] in case @f is contained in the dag subdi¤erential @ y f de…ned there. We use the conditions:

(C) if for some w; x 2 C; f is constant on [w; x], then one has hx ; w xi 0 for some x 2 @f (x):

(M) if x is a local minimizer of f; then 0 2 @f (x). Condition (C) is satis…ed when @f (x) \ @ D f (x) 6 = ?;
where @ D f is the Dini-Hadamard subdi¤erential of f; in particular when f is Gâteaux-di¤erentiable at each point of C and f 0 (x) 2 @f (x): Condition (M) is a natural condition which is satis…ed by all sensible generalizations of the derivative used for minimization problems.

Let us recall that a function f is radially nonconstant if one cannot …nd any line segment on which f is constant.

The …rst assertion of the following proposition can be given when f is a radially continuous l.s.c. function and @f is a valuable generalized di¤erential since if f is @f -pseudoconvex, then it is @f -quasiconvex and, by Proposition 4, it is quasiconvex; see also in [1, Theorem 4.1] for the case X admits a smooth renorming. A comparison of assertions (b) and (c) with [1, Proposition 5.2] is made in the example below. The proof of the last assertion is similar to the one of [1, Proposition 5.2]; note that one does not need that @ be valuable.

Proposition 6 Let C be convex, let f : C ! R 1 be @f -pseudoconvex and @f (x) 6 = ? for all x 2 C. Then (a) f is quasiconvex. (b) If f satis…es condition (C), then it is semi-strictly quasiconvex. (c) If f satis…es condition (M)
, then every local minimizer is a global minimizer. Moreover, if f is radially continuous and l.s.c., then it is semi-strictly quasiconvex.

(d) If f is radially nonconstant, then f is strictly @f -pseudoconvex.

Proof. (a) Let f be @f -pseudoconvex. Suppose that f is not quasiconvex. Then there exist w; x; y 2 C such that x 2 (w; y) and f (x) > f (w); f (x) > f (y): By @f -pseudoconvexity of f; for all x 2 @f (x) one has hx ; w xi < 0 and hx ; y xi < 0, a contradiction with the linearity of x :

(b) Now, suppose that f is @f -pseudoconvex and satis…es condition (C). If f is not semi-strictly quasiconvex then there exist w; x; y 2 C with x 2 (w; y) such that f (w) < f (y) f (x): Thus, by (a), f (x) = f (y) and x is a maximizer of f on [w; x]: Now, if f ([x; y]) is not constant then there exists z 2 (x; y) such that f (z) < f (x), a contradition with the quasiconvexity of f and f (w) < f (x). Hence, f ([x; y]) is constant. Since f is @fpseudoconvex and f (w) < f (x), one has hx ; w xi < 0 for all x 2 @f (x): On the other hand, by condition (C), there exists x 2 @f (x) such that 0 hx ; y xi: a contradiction.

(c) The …rst assertion follows from the de…nition of @f -pseudoconvexity of f and condition (M). For the second assertion, see [START_REF] Hadjisavvas | Handbook of generalized convexity and generalized monotonicity[END_REF]Prop. 11.5 (iii)]. Note that for this implication, one does not need that @ be valuable.

(d) See [1, Proposition 5.2]. Example 5. Let f : R ! R be given by f (x) = 0 for x 0 and f (x) = x for x > 0: Since f is not radially nonconstant, we cannot use [1, Proposition 5.2]. However, appropriate choices of @f allow to apply Proposition 6:

(a) Let @f (x) = R for x < 0 and @f (x) = P for x 0: Then f is @f -pseudoconvex and condition (C) is satis…ed. Thus, f is semi-strictly quasiconvex. Note that condition (M) is not satis…ed since 0 is a local minimizer of f but 0 = 2 @f (0): (b) If @f (x) = R for x 0 and @f (x) = P for x > 0; then condition (M) is satis…ed. A simple stability property is given in the following lemma (here the convexity of C is not needed).

Lemma 7 Let I be a …nite set. For i 2 I, let

f i : C ! R 1 ; let @f i : C X and let f := sup i2I f i . For x 2 C; let I(x) := fi 2 I : f i (x) = f (x)g and @f (x) := co [ i2I(x) @f i (x).
If for i 2 I(x), f i is @f i -pseudoconvex (resp. @f i -protoconvex, @f i -quasiconvex) at x, then f is @fpseudoconvex (resp. @f -protoconvex, @f -quasiconvex) at x: Proof. Let w 2 C be such that f (w) < f (x). Then for all i 2 I(x) one has f i (w) < f i (x): Since f i is @f i -pseudoconvex, for all x 2 [ i2I(x) @f i (x); one has hx ; w xi < 0: Now, let x 2 @f (x): Then there exist t i 2 [0; 1], P i2I(x) t i = 1 and x i 2 @f i (x) for i 2 I(x) such that x = P i2I(x) t i x i ; hence hx ; w xi < 0. A similar proof can be given to show that f is @f -protoconvex or @f -quasiconvex at x.

Links with previous works

In [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF], assuming a generalized directional derivative h : V C ! R of f is given, we de…ned concepts which seem to be related to the present notions. Let us make a precise comparison. Recall that

f is h-pseudoconvex (resp. h-quasiconvex, h-protoconvex ) at x if for all w 2 C f (w) < f (x) ) h(x; w x) < 0 (resp. f (w) < f (x) ) h(x; w x) 0; resp. f (w) f (x) ) h(x; w x) 0).
Let us …rst consider the case we dispose of a general di¤erential @f which is a generalized pseudo-di¤erential of f: The concept of pseudo-di¤erential of f has been introduced by Jeyakumar and Luc in [START_REF] Jeyakuma | Nonsmooth vector functions and continuous optimization[END_REF] when h + and h + are the Dini derivatives of f . We introduce a slight variant which enables to encompass [14, De…nitions 2.1, 2.2] and [START_REF] Jeyakuma | Nonsmooth vector functions and continuous optimization[END_REF].

De…nition 8 Let f : C ! R and h : V C ! R: We say that a subset @f (x) of X is:
(a) an upper h-pseudo-di¤ erential of f at x if for all (x; u) 2 V C; we have h(x; u) supfhx ; ui : x 2 @f (x)g:

(b) a lower h-pseudo-di¤ erential of f at x if for all (x; u) 2 V C; we have h(x; u) inffhx ; ui : x 2 @f (x)g:

(c) an (h + ; h + )-pseudo-di¤ erential of f at x; where h + ; h + : V C ! R; if it is an upper h + -pseudo di¤ erential of f at x and a lower h + -pseudo di¤ erential of f at x. We say that @f (x) is upper exact (resp. lower exact) if for all u 2 X one has supfhx ; ui : x 2 @f (x)g = maxfhx ; ui : x 2 @f (x)g (resp. inffhx ; ui : x 2 @f (x)g = minfhx ; ui : x 2 @f (x)g) whenever the supremum (resp. in…mum) is …nite. Clearly, if @f (x) is weak compact then @f (x) is upper exact and lower exact.

Proposition 9 Let f : C ! R 1 , h : V C ! R and let @f (x) be an upper h-pseudo-di¤ erential of f at x. If f is @f -quasiconvex (resp. @f -protoconvex) at x; then f is h-quasiconvex (resp. h-protoconvex) at x. If f is @f -pseudoconvex at x and if @f (x) is upper exact, then f is h-pseudoconvex at x:
Proof. Let f be @f -quasiconvex (resp. @f -protoconvex) at x and let w 2 C be such that f (w) < f (x) (resp. f (w) f (x)). Since h(x; w x) supfhx ; w xi : x 2 @f (x)g 0; f is h-quasiconvex (resp. h-protoconvex) at x. Now let f be @f -pseudoconvex at x and let w 2 [f < f (x)]: Then for all x 2 @f (x); one has hx ; w xi < 0: Thus, if @f (x) upper exact, one has maxfhx ; ui : x 2 @f (x)g < 0; hence h(x; w x) < 0 and so f is hpseudoconvex at x:

We have a kind of converse.

Proposition 10 Let @f (x) be a lower h-pseudo-di¤ erential of f at x. If f is h-pseudoconvex then f is eventually @f -pseudoconvex. If f is h-quasiconvex (resp. h-protoconvex) at x and if @f (x) is lower exact, then f is eventually @f - quasiconvex (resp. eventually @f -protoconvex) at x.
Proof. Since f is h-pseudoconvex at x, for any w 2 C; f (w) < f (x), one has 0 > h(x; w x) inffhx ; w xi : x 2 @f (x)g: Hence, there exists x 2 @f (x) such that hx ; w xi < 0: f is eventually @f -pseudoconvex at x. Now, let f be h-quasiconvex (resp. h-protoconvex) at x. For any w 2 C; w 2 [f < f (x)] (resp. w 2 [f f (x)]), one has 0 h(x; w x) inffhx ; w xi : x 2 @f (x)g:

Since @f (x) is lower exact, there exists x 2 @f (x) such that hx ; w xi 0: Hence, f is eventually @fquasiconvex (resp. eventually @f -protoconvex) at x. Now, let us assume that @f is the generalized di¤erential of f associated with a bifunction h in the following way:

@f (x) := @ h f (x) := @h(x; )(0) := fx 2 X : 8u 2 X; hx ; ui h(x; u)g:

We will use the following conditions on h : (S) 8w; x 2 C h(x; w x) = supfhx ; w xi : x 2 @ h f (x)g:

(S') 8w; x 2 C h(x; w x) = maxfhx ; w xi : x 2 @ h f (x)g:
Proposition 11 (a) If f is h-pseudoconvex (resp. h-quasiconvex, h-protoconvex) at x; then f is @ h f -pseudoconvex (resp. @ h f -quasiconvex, @ h f -protoconvex) at x. (b) Conversely, if f is @ h f -pseudoconvex at x and if h satis…es condition (S'), then f is h-pseudoconvex at x. (c) If f is @ h f -quasiconvex (resp. @ h f -protoconvex) at x and if h satis…es condition (S), then f is h- quasiconvex (resp. h-protoconvex) at x. Proof. (a) If f is h-pseudoconvex at x; then for w 2 [f < f (x)] and x 2 @ h f (x) one has hx ; w xi h(x; w x) < 0: Thus f is @ h f -pseudoconvex. Now, if f is h-quasiconvex (resp. h-protoconvex) at x then for w 2 [f < f (x)] (resp. w 2 [f f (x)]
), x 2 @ h f (x); one has hx ; w xi h(x; w x) 0 and then f is @ h f -quasiconvex (resp. @ h f -protoconvex) at x.

(b) It follows from Proposition 9, since when h satis…es condition (S'), @ h f (x) is an upper h-pseudo-di¤erential of f at x which is upper exact.

(c) Since h satisfy condition (S), @ h f is an upper h-pseudo-di¤erential of f at x: Thus, the conclusion follows from the …rst assertion of Proposition 9.

Optimality conditions for problems with constraints

In the present section, we apply the preceding concepts to the constrained minimization problem

(C) min f (x) subject to x 2 C;
where C is a subset of X; and f : C ! R 1 . Let S be the set of solutions to (C) and let @f : C X be a generalized di¤erential of f:

Let us give a su¢ cient condition for (C).

Proposition 12 (a) Let f be @f -pseudoconvex at a 2 C. If 0 2 @f (a) + N (C; a); then a is a solution to (C).

(b) Let f be eventually @f -pseudoconvex at a 2 C. If @f (a) N (C; a); then a is a solution to (C):

(c) If f is @f -pseudoconvex at a, if @f (a)
is upper exact and if the following condition holds, then a is a solution to (C) : sup

a 2@f (a) ha ; ui 0 8u 2 V (C; a): (10)
Proof. (a) Let us note that the assumption of (a) is stronger than the assumption of (c). However, we give a direct proof. If 0 2 @f (a) + N (C; a); then there exists a 2 @f (a) such that ha ; x ai 0 for all x 2 C: Since f is @f -pseudoconvex at a, one has f (x) f (a) for all x 2 C : a 2 S:

(b) Suppose @f (a) N (C; a) and f is eventually @f -pseudoconvex at a: If for some x 2 C one has f (x) < f (a) then there exists a 2 @f (a) such that ha ; x ai < 0 , a contradiction with a 2 N (C; a):

(c) Supposing that a = 2 S we show that (10) does not hold. Let x 2 C be such that f (x) < f (a): Since f is @f -pseudoconvex at a; for all a 2 @f (a); we have ha ; x ai < 0: Since @f (a) is upper exact, we get maxfha ; x ai : a 2 @f (a) < 0 and (10) does not hold.

It follows from Lemma 2 that the preceding proposition implies the result in [21, Prop. 5].

The following example shows that with some suitable choices of @f , the assumptions of the preceding proposition reduce to the condition 0 2 @f (a) + N (C; a): Example 6. Let us take for @f the subdi¤erentials @ 0 f and @ ^f given in [30, Example 7.2] as follows

x 2 @ 0 f (x) , f (x) = min hx ;wi>1 f (w); hx ; xi > 1; x 2 @ ^f (x) , f (x) = min hx ;wi 1 f (w); hx ; xi 1: Then, any function f is @f -pseudoconvex: if x 2 @ 0 f (x) is such that hx ; w xi 0; then hx ; wi hx ; xi > 1, hence f (w) f (x). Similarly, we have that f is @ ^f -pseudoconvex.
The next proposition is similar to [START_REF] Aussel | Adjusted sublevel sets, normal operator, and quasi-convex programming[END_REF]Prop. 4.1]. Note that here, we do not need the quasiconvexity of f and moreover @f (x) may be di¤erent from N < f (x)nf0g: Also, we do not need the additional assumptions of Proposition 5 which would enable to apply the preceding proposition.

Proposition 13 Let C be convex and let f be @f -quasiconvex and radially upper semicontinuous at a 2 C. Assume that 0 = 2 @f (a) and C ? := fx 2 X : 8w; x 2 C; hx ; xi = hx ; wig = f0g: If 0 2 @f (a) + N (C; a); then a is a solution to (C).

Proof. Since 0 2 @f (a) + N (C; a), there exists a 2 @f (a) such that ha ; x ai 0 for all x 2 C: Since C ? = 0; there exists w 2 C such that ha ; w ai > 0: Now, for any given x 2 C; t 2 (0; 1); we de…ne x t := (1 t)x + tw so that ha ; x t ai = (1 t)ha ; x ai + tha ; w ai > 0 Since f is @f -quasiconvex at a; one has f (x t ) f (a) and then f (x) f (a) by radial upper semicontinuity of f at a: Thus, a 2 S:

When a suitable bifunction h is available and @f is an upper h-pseudo-di¤erential of f , condition [START_REF] Daniilidis | An appropriate subdi¤ erential for quasiconvex functions[END_REF] turn to be necessary. Proposition 14 Let C be convex, h : V C ! R be such that h(a; ) D + f (a; ) and let a subset @f (a) of X be an upper h-pseudo-di¤ erential of f at a. If a 2 S then condition (10) holds.

Proof. Let a 2 S and u 2 V (C; a). Let x 2 C be such that u := x a: Then 0 D + f (a; x a) h(a; x a) supfha ; x aia 2 @f (a)g: Under some additional convexity assumptions, the su¢ cient condition of Proposition 12 (a) also is necessary.

Proposition 15 Let C be convex, let h : V C ! R be such that h(a; ) D + f (a; ) and let @f (a) be an upper h-pseudo-di¤ erential of f at a. Assume that h(a; ) is sublinear and continuous. If a 2 S then 0 2 w cl(co(@f (a))) + N (C; a): In particular, if @f (a) is weak closed convex, one has 0 2 @f (a) + N (C; a).

Note that the assumption that @f (a) is weak closed convex is satis…ed when @f (a) = @ h f (a): Proof. Let a 2 S: One has h(a; v) 0 for all v 2 V (C; a) by [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF]Proposition 27] or, equivalently, h(a; )+ V (C;a) ( ) 0: Since C is convex, V (C; a) is a convex set. Hence we have 0 2 @(h(a; )+ V (C;a) ( ))(0) so that 0 2 @h(a; )(0)+@ V (C;a) (0). Now since @f (a) is an upper h-pseudo-di¤erential of f at a; the set @h(a; )(0) is contained in the weak closed convex hull of @f (a); which is @f (a). Hence, we get 0 2 @f (a) + N (C; a).

Let us give some other optimality conditions in the spirit of Minty variational inequalities and [START_REF] Aussel | Adjusted sublevel sets, normal operator, and quasi-convex programming[END_REF]. Recall ( [START_REF] Aussel | On quasivariational inequalities[END_REF]) that a multimap T : C X is said to be upper sign-continuous on C if, for every w; x 2 C; the following implication (in which x t := tx + (1 t)w) holds: 8t 2 (0; 1); inffhx t ; w xi : x t 2 T (x t )g 0 ) supfhx ; w xi : x 2 T (x)g 0:

We will use the following assumption taken from ( [START_REF] Aussel | On quasivariational inequalities[END_REF]):

(A) for every x 2 C there exists a convex neighborhood V x of x and an upper sign-continuous operator T : V x \ C ! 2 X with nonempty values satisfying T (w) @f (w) for all w 2 V x \ C.

Proposition 16 Let C be convex, let @f : C X and let f be @f -protoconvex on C. If a 2 S, then for any x 2 C one has inffhx ; x ai : x 2 @f (x)g 0:

If assumption (A) holds then condition (10) is satis…ed. If moreover T (a) is convex, w*-compact, then 0 2 @f (a) + N (C; a):

Proof. Suppose that there exist x 2 C and x 2 @f (x) such that hx ; x ai < 0. Since f is @f -protoconvex at x, one has f (a) > f (x): a contradition with a 2 S: Thus, inffhx ; x ai : x 2 @f (x)g 0 for all x 2 C: Since, for every x 2 C; there exists a convex neighborhood V x of x such that for any w 2 V x \ C, T (w) @f (w), one has inf w2Vx\C inffhw ; w ai : w 2 T (w)g 0:

Since T is upper sign-continuous and V x \ C is a convex set, for w 2 V x \ C and t 2 (0; 1); we have w t := (1 t)a + tw 2 V x \ C and inffhw t ; w t ai : w t 2 T (w t )g 0 hence inffhw t ; w ai : w t 2 T (w t )g 0: By upper sign-continuity of T; we deduce that supfha ; w ai : a 2 T (a)g 0 for all w 2 V x \ C: Using the convexity of C; we get condition [START_REF] Daniilidis | An appropriate subdi¤ erential for quasiconvex functions[END_REF] Under special assumptions, one has a stronger necessary condition.

Proposition 18 (a) If f is ( @f )-protoconvex at a 2 S; then @f (a) N (C; a): (b) If f is eventually ( @f )-protoconvex at a 2 S; then condition (10) is satis…ed.

Proof. (a) Let a 2 S: Since f is @f -protoconvex at a and, for all x 2 C; f (x) f (a); for all a 2 @f (a) one has h a ; x ai 0 for all x 2 C so that a 2 N (C; a). Thus, @f (a) N (C; a):

(b) Let a 2 S: Since f is eventually @f -protoconvex at a and, for all x 2 C; f (x) f (a); there exists a 2 @f (a) such that ha ; x ai 0. Thus, we get inf x2C max a 2@f (a) ha ; x ai 0:

With a similar proof, we have the following proposition.

Proposition 19 Let a 2 C be a strict solution of (C) (i.e. for all x 2 C; f (a) < f (x)).

(a) If f is ( @f )-quasiconvex at a; then @f (a) N (C; a): (b) If f is ( @f )-pseudoconvex at a; then @f (a) N (C; a)nf0g:

Example 6 shows that if we take @f := @ 0 ( f ) or @f := @ ^( f ); the pseudoconvexity assumption of the preceding proposition is automatically satis…ed.

Mathematical programming problems

Let us consider now the case in which the constraint set C is de…ned by a …nite family of inequalities, so that problem (C) turns into the mathematical programming problem (M) min f (x) subject to x 2 C := fx 2 X : g 1 (x) 0; :::; g n (x) 0g;

where f : W ! R 1 , g i : W ! R 1 and W is a subset of X.

Let a 2 C, and let I := f1; :::; ng; I(a) := fi 2 I : g i (a) = 0g.

Let us …rst present su¢ cient optimality conditions.

Proposition 20 Let f be @f -pseudoconvex at a 2 C and g i be @g i -protoconvex at a for i 2 I(a). If for all x 2 C, there exist a 2 @f (a), a i 2 @g i (a) (i 2 I(a)) and (y i ) 2 R I(a) + such that ha + P i2I(a) y i a i ; x ai 0 (in particular, if a + P i2I(a) y i a i = 0 for some a 2 @f (a) and some a i 2 @g i (a) for i 2 I(a)), then a is a solution to problem (M).

Proof. Suppose on the contrary that there exists some x 2 C such that f (x) < f (a): Since f is @fpseudoconvex at a and a 2 @f (a); we have ha ; x ai < 0. Since for i 2 I(a), g i is @g i -protoconvex at a and a i 2 @g i (a), we have ha i ; x ai 0. Taking (y i ) 2 R I(a) + as in our assumption, we get a contradiction with the relation ha ; x ai + X i2I(a) y i ha i ; x ai < 0 deduced from the preceding inequalities. Remark (a) According to Proposition 4, the condition that g i is @g i -protoconvex at a can be replaced by quasiconvexity of g i and @g i (a) @ r g i (a) for i 2 I(a). (b) By Lemma 2 (b) and (c), the preceding proposition implies [START_REF] Linh | Optimality conditions for quasiconvex programming[END_REF]Thm. 11].

A variant with a similar proof can be given.

Proposition 21 Let f be eventually @f -pseudoconvex at a 2 C and let g i be eventually @g i -protoconvex at a for i 2 I(a). If for all x 2 C, a 2 @f (a) and a i 2 @g i (a) (i 2 I(a)) there exists (y i ) 2 R I(a) + such that ha + P i2I(a) y i a i ; x ai 0, then a is a solution to problem (M).

  Condition (S) (resp. (S')) is satis…ed when C is open and for all x 2 C the function h(x; ) is sublinear and l.s.c. (resp. continuous).

  Moreover, when T (a) is convex, weak* compact, one has inf x2C max a 2T (a) ha ; x ai 0: Now, applying the Sion minimax theorem or [3, Lemma 2.1(iii)], we get max a 2T (a) inf x2C fha ; x aig 0: Hence there exists some a 2 T (a) @f (a) such that a 2 N (C; a): Now let us turn to optimality conditions formulated in terms of normal cones to sublevel sets. The following result can be couched in terms of the adjusted normal cone to f as in [5, Prop. 5.1]; here we give a short proof which avoids this concept. Proposition 17 Let f be a continuous, semi-strictly quasiconvex function on a convex C and let a 2 X be such that f (a) > inf X f: Then a 2 S if, and only if, 0 2 N f (a)nf0g + N (C; a): Proof. Since a 2 S we have that S < f (a) is disjoint from C: Since these two sets are convex and S < f (a) is open, the Hahn-Banach theorem yields some a 2 X nf0g and some 2 R such that 8w 2 S < f (a); 8x 2 C ha ; wi > ha ; xi: Since f is semi-strictly quasiconvex, a belongs to C and the closure of S < f (a): Thus ha ; ai = and 8w 2 S < f (a); 8x 2 C ha ; w ai > 0 ha ; x ai: Therefore a 2 N (C; a) and a 2 N f (a)nf0g since N < f (a) = N f (a) by Lemma 3. Now, assume that 0 2 N f (a)nf0g + N (C; a): By Lemma 3, f is N f nf0g-pseudoconvex at a: Hence, we get a 2 S by Proposition 12(a).

Another su¢ cient condition can be given.

Proposition 22 Let f be @f -quasiconvex and radially upper semicontinuous at a 2 C and let g i be quasiconvex for i 2 I: Assume that 0 = 2 @f (a), C ? = f0g and @g i (a) @ r g i (a) for i 2 I(a). If there exists (y i ) 2 R

y i a i = 0 for some a 2 @f (a) and some a i 2 @g i (a) (i 2 I(a)), then a is a solution to problem (M).

Proof. By Proposition 4 (a), g i is @g i -protoconvex at a: Let a 2 @f (a), a i 2 @g i (a) and

y i a i = 0: Since for all x 2 C, i 2 I(a); one has g i (x) g i (a); hence ha i ; x ai 0; we obtain a = P i2I(a)

y i a i N (C; a): Then Proposition 13 yields the conclusion. Now let us turn to necessary optimality conditions for (M). Let us …rst consider the case of a single constraint which is quasiconvex.

Proposition 23 Let @f : W X and let f be @f -protoconvex. Let g 1 be quasiconvex, a 2 S and g 1 (a) = 0. If assumption (A) holds and if T (a) is convex w*-compact, then there exists y 1 2 R + such that Now, let us turn to the general case. We shall use the following lemma in which a) . Its proof is a consequence of the rule for computing the subdi¤erential of a sum of convex functions, applied to the case the functions are the indicator functions of the sets C i := g 1 i (( 1; 0]):

Lemma 24 Assume g i is quasiconvex for all i 2 I, g i is u.s.c. at a for all i 2 InI(a) and let C i := g 1 i (( 1; 0]), a 2 X. Assume that one of the following two conditions holds: (a) there exist some k 2 I(a) and some z 2 C k such that g i (z) < 0 for each i 2 I(a)nfkg (Slater condition). (b) X is complete, g i is l.s.c. for all i 2 I(a) and R + ( Q

This lemma and Proposition 16 entail the following result.

Proposition 25 Let @f : W X and let f be @f -protoconvex. Suppose g 1 ; :::; g n satisfy the assumptions of the preceding lemma, assumption (A) holds and T (a) is convex w*-compact. Then there exists

y i N (S gi (a); a):

Note that this optimality condition can be formulated in terms of Gutiérrez functions to the constraints. Recall from [START_REF] Linh | Optimality conditions for quasiconvex programming[END_REF] that f is a Gutiérrez function at a if its sublevel set S f (a) is convex and such that

By the preceding proposition and [21, Lemma 9], if the constraints (g i ) i2I(a) are Gutiérrez function at a in the preceding proposition as in the preceding lemma, then there exists (y i ) 2 R I(a) + such that 0 2 @f (a) + P i2I(a) y i @ g i (a):

Using representations of normal vectors to sublevels sets such as [START_REF] Borwein | Techniques of Variational Analysis[END_REF]Theorem 3.3.4], [START_REF] Ngai | A fuzzy necessary optimality condition for non-lipchitz optimization in Asplund spaces[END_REF]Theorem 4.1], [START_REF] Penot | Harmonic sums and duality[END_REF]Proposition 5.4] under appropriate assumptions, one can derive from the last proposition necessary conditions in fuzzy or limiting forms.


N T H Linh

J.-P Penot

Generalized A¢ ne Functions and Generalized Di¤erentials

Keywords: colinvex, colin…ne, generalized di¤erential, optimization problem, protoconvex function, pseudoconvex function, pseudolinear function, quasiconvex function. Mathematics Subject Classi…cation: 26B25, 46G05, 49K27, 90C26, 90C32

We study some classes of generalized a¢ ne functions, using a generalized di¤erential. We study some properties and characterizations of these classes and we devise some characterizations of solution sets of optimization problems involving such functions or functions of related classes.

Introduction

A generalized a¢ ne function is a function which is both generalized convex and generalized concave. Such functions have been studied in [START_REF] Aggarwal | Pseudolinearity and e¢ ciency via Dini derivatives[END_REF], [START_REF] Cambini | On generalized linearity of quadratic fractional functions[END_REF], [START_REF] Cambini | Generalized convexity and optimality conditions in scalar and vector optimization[END_REF], [START_REF] Chew | Pseudolinearity and e¢ ciency[END_REF], [START_REF] Jeyakuma | On characterizing the solution sets of pseudolinear programs[END_REF], [START_REF] Komlosi | First and second characterizations of pseudolinear functions[END_REF], [START_REF] Kortanek | Pseudoconcave programming and Lagrange regularity[END_REF], [START_REF] Kruk | Pseudolinear programming[END_REF], [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF], [START_REF] Martinez-Legaz | Quasiconvex duality theory by generalized conjugation methods[END_REF], [START_REF] Martos | Nonlinear programming: Theory and methods[END_REF], [START_REF] Rapcsak | On pseudolinear functions[END_REF], [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF]. Among them are quasia¢ ne functions, i.e. those functions which are both quasiconvex and quasiconcave (see [START_REF] Martinez-Legaz | Quasiconvex duality theory by generalized conjugation methods[END_REF], [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF]). Also, among them are pseudoa¢ ne functions, i.e. those functions which are di¤erentiable, pseudoconvex and pseudoconcave, also called pseudolinear functions (see [START_REF] Cambini | Generalized convexity and optimality conditions in scalar and vector optimization[END_REF], [START_REF] Komlosi | First and second characterizations of pseudolinear functions[END_REF], [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF]). Given a bifunction h; the class of h-colin…ne functions also belongs to this category (see [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF]). These references provide interesting, non trivial examples of generalized a¢ ne functions; in particular, fractional functions are noticeable pseudo-a¢ ne functions and quadratic pseudo-a¢ ne functions can be characterized.

It is the purpose of the present paper to introduce and study new concepts of generalized a¢ ne functions, as it has been done for generalized convex functions in [START_REF] Linh | Generalized convex functions and generalized di¤ erentials[END_REF]. Here, to de…ne these classes, we use a generalized di¤erential, i.e. a set-valued map @f; as a substitute for the derivative of f: This concept allows much ‡exibility as it encompasses several notions of nonsmooth analysis. We give some elementary properties and characterizations for these generalized a¢ ne functions. We also present applications to the characterization of the solution set of a constrained minimization problem.

Notation and de…nitions

Throughout this paper, X is a normed vector space (n.v.s.), X is the dual space of X, C is a nonempty subset of X, and f : C ! R. We assume that a set-valued map @f : C X is given which stands for a substitute to the derivative of f ; we call it a generalized di¤ erential of f . As observed in [START_REF] Linh | Generalized convex functions and generalized di¤ erentials[END_REF], the choice for @f is not limited to the subdi¤erentials of nonsmooth analysis; one can also take the convexi…cators of [START_REF] Demyanov | Contructive nonsmooth analysis[END_REF], the pseudo-di¤erentials of Jeyakumar and Luc ( [START_REF] Jeyakuma | Nonsmooth vector functions and continuous optimization[END_REF]), and much more. We assume that @f (x) 6 = ? for all x 2 C although such an assumption could be relaxed for several results. We denote by P (resp. R + ) the set of positive numbers (resp. non negative numbers) and R (resp. R 1 ) stands for R [ f 1; +1g (resp. R [ f+1g).

We recall that the visibility cone V (C; x) of C at x 2 C is the cone generated by C x:

V (C; x) := P(C x) := fr(c x) : r 2 P; c 2 Cg:

Laboratoire de Mathématiques, CNRS UMR 5142, Faculté des Sciences, Av. de l'Université 64000 Pau, France (honglinh98t1@yahoo.com) y Laboratoire de Mathématiques, CNRS UMR 5142, Faculté des Sciences, Av. de l'Université 64000 Pau, France (jeanpaul.penot@univ-pau.fr)

The visibility bundle of C is the set V C := f(x; u) 2 C X : 9r 2 P; w 2 C; u = r(w x)g = [ x2C fxg V (C; x):

It contains the radial tangent bundle of C which is the set

T r C := f(x; u) 2 C X : 9 (r n ) ! 0 + ; x + r n u 2 C 8ng = [ x2C fxg T r (C; x):
We also use the tangent bundle of C which is the set

T C := f(x; u) 2 C X : 9(r n ) 2 P N ; (u n ) ! u; x + r n u n 2 C 8ng = [ x2C fxg T (C; x):
The upper and the lower radial derivatives (or upper and lower Dini derivatives) of f at x 2 C in the direction u 2 T r (C; x), are de…ned by

D + f (x; u) = lim sup t!0+; x+tu2C 1 t [f (x + tu) f (x)]; D + f (x; u) = lim inf t!0+; x+tu2C 1 t [f (x + tu) f (x)]:
We recall the following de…nitions (see [START_REF] Linh | Generalized convex functions and generalized di¤ erentials[END_REF][START_REF] Penot | Are generalized derivatives useful for generalized convex function[END_REF][START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF]).

De…nition 2.1 Let f : C X ! R. A function f is said to be (a) @f -pseudoconvex at x if for all w 2 C :

f (w) < f (x) ) for all x 2 @f (x) : hx ; w xi < 0: (1)
(b) @f -quasiconvex at x if for all w 2 C :

f (w) < f (x) ) for all x 2 @f (x) : hx ; w xi 0: (2)
(c) @f -protoconvex at x if for all w 2 C :

f (w) f (x) ) for all x 2 @f (x) : hx ; w xi 0: (3)
We add the word "eventually" when in the preceding implications "for all" is changed into "there exists". Thus, for instance, f is eventually @f -pseudoconvex at x if for all w 2 C satisfying f (w) < f (x) there exists x 2 @f (x) such that hx ; w xi < 0:

Clearly

f is (eventually) @f -pseudoconvex at x =) f is (eventually) @f -quasiconvex at x; (4) f is (eventually) @f -protoconvex at x =) f is (eventually) @f -quasiconvex at x: (5)
Now we introduce some de…nitions related to generalized concavity .

De…nition 2.2 Let f : C ! R, g := f and some @g : C X be given. The function f is said to be (eventually) @( f )-pseudoconcave (resp. (eventually) @( f )-quasiconcave, (eventually) @( f )-protoconcave) at x if for g := f; the function g is (eventually) @g-pseudoconvex (resp. (eventually) @g-quasiconvex, (eventually) @g-protoconvex) at x. A function f : C ! R is said to be (eventually) @f -pseudoa¢ ne if it is both (eventually) @f -pseudoconvex and (eventually) @( f )-pseudoconcave. It is said to be @f -protoa¢ ne if it is @f -protoconvex and @( f )-protoconcave.

Remark 2.1 (a) If we assume that @( f

)(x) = @f (x), then f is @( f )-pseudoconcave (resp. @( f )- quasiconcave, @( f )-protoconcave) at x if and only if f is @f -pseudoconvex (resp. @f -quasiconvex, @f -protoconvex) at x. (b) When @f (x) := D f (x) [ D f (x)
where D f (x) and D f (x) is a subdi¤ erential of f and f at x respectively, one has @( f )(x) = @f (x).

We recall from [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF] that f : C ! R is said to be h-colin…ne at x 2 C; where h : V C ! R, if there exists : C C ! P such that for every w 2 X;

f (w) f (x) = (w; x)h(x; w x):
Here we introduce a concept in which the bifunction h is replaced by a generalized di¤erential @f .

De…nition 2.3 (a) A function f : C X ! R is @f -colin…ne at x 2 C if there exists : C Graph(@f ) ! P such that for every w 2 C; for any x 2 @f (x) f (w) f (x) = (w; x; x )hx ; w xi: (6) (b) A function f : C X ! R is eventually @f -colin…ne at x 2 C
if there exists : C Graph(@f ) ! P such that for every w 2 C; there exists x 2 @f (x) satisfying [START_REF] Chew | Pseudolinearity and e¢ ciency[END_REF].

A function f is said to be @f -colin…ne (resp. @f -pseudoconvex...) on C if it is @f -colin…ne (resp. @fpseudoconvex...) at each point of C: For this reason we have chosen to consider is de…ned on C Graph(@f ) in De…nition 2.3 rather than on C @f (x) although only the values of on C @f (x) play a role. Similarly, we have chosen to write De…nition 2.3(b) in that way in order to stress the analogy with (a). Note that this de…nition can be rephrased as: f is eventually @f -colin…ne if there exists : C C ! P and g : C C ! X such that for every w; x 2 X; g(w; x) 2 @f (x) and f (w) f (x) = (w; x)hg(w; x); w xi: When f is di¤erential and @f := fDf g; f is @f -colin…ne if, and only if, f is pseudoa¢ ne.

Although the preceding de…nitions are quite restrictive, they are satis…ed in some cases of signi…cant interest.

Note that if f is a h-colin…ne function, where h : V C ! R is linear and continuous in its second variable, with `(x) := h(x; ) for all x 2 C; then f is a @f -colin…ne function for @f := f`g and f is an eventually @f -colin…ne function when @f f`g: Hence, every pseudoa¢ ne (di¤erentiable) function is @f -colin…ne where @f := fDf g or eventually @f -colin…ne where @f fDf g: Remark that the converse is not true in general, as shown in Example 2.3, where f is eventually @f -colin…ne but f is not quasiconvex, hence f is not pseudoa¢ ne.

Example 2.1 Let X = R n and let a; b 2 X ; ; 2 R. For C := fx 2 X : bx + > 0g; let f : C ! R be given by f (x) = ax + bx + for all x 2 C. When @f (x) := frf (x)g; f is @f -colin…ne at x since rf (x) = (bx + )a + (ax + )b (bx + ) 2 ; f (w) f (x) = bx + bw + rf (x)(w x): If @f (x) frf (x)g then f is eventually @f -colin…ne at x.
Example 2.2 Let the function f : R ! R be given by f (x) := x 3 . Then f is not pseudoa¢ ne. But if @f := P or ? 6 = @f P, then f is @f -colin…ne. If @f \ P 6 = ? then f is eventually @f -colin…ne.

Example 2.3 Let f : R ! R be given by f (x) = 0 for x 6 = 0 and f (0) = 1: Let @f (x) = f 1; 0; 1g for x 6 = 0 and @f (0) = f 1; 1g: Then f is eventually @f -colin…ne but f is not quasiconvex.

Example 2.4 Let f : P ! R be given by f (x) =

x 2 1 x : Then rf (x) = x 2 + 1 x 2 : If for all x 2 P, @f (x) = rf (x) or @f (x) P then f is @f -colin…ne. If @f (x) \ P 6 = ? for all x 2 P, then f is eventually @f -colin…ne.

Example 2.5 Let X be a n.v.s. and let g : X ! R be @g-colin…ne. For C := fx 2 X : g(x) > 0g; let f (x) = p g(x) and @f P@g. f is @f -colin…ne on C since for any w; x 2 C and x f 2 @f (x), there exist r 2 P and x g 2 @g(x) such that x f = rx g and

hx f ; w xi = rhx g ; w xi = r g(w) g(x) (w; x; x g ) = r( p g(w) + p g(x)) f (w) f (x) (w; x; x g ) := f (w) f (x) f (w; x; x f ) :
Similarly, if @f \ P@g 6 = ? then f is eventually @f -colin…ne on C.

Example 2.6 More generally, let X be a n.v.s., g : X ! R be @g-colin…ne on C := fx 2 X : g(x) > 0g:

Then, for any p 2 N, f (:) := g p (:) := (g(:)) p and @f P@g, f is @f -colin…ne on C since for any w; x 2 C and x f 2 @f (x), there exist r 2 P and x g 2 @g(x) such that x f = rx g and

f (w) f (x) = g p (w) g p (x) = (g(w) g(x))( p 1 X k=0 g k (w)g p k 1 (x)) = g (w; x; x g )hx g ; w xi( p 1 X k=0 g k (w)g p k 1 (x)) = f (w; x; x f )hx f ; w xi where f (w; x; x f ) := r 1 g (w; x; x g )( p 1 P k=0 g k (w)g p k 1 (x)): Similarly, if @f \ P@g 6 = ? then f is eventually @f -colin…ne on C.
The following result is similar to [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF]Prop. 4], but instead of using a bifunction as a generalized directional derivative, we use an arbitrary generalized di¤erential. We use a terminology similar to the one in [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF]Prop. 4] in view of the fact that the vector v := (w; x; x )(w x) in the invexity relation [START_REF] Demyanov | Contructive nonsmooth analysis[END_REF] is colinear to w x; a feature close to what occurs in the convex case.

Proposition 2.1 For any function f : C ! R, x 2 domf and any multimap @f : C X , the following assertions are equivalent.

(a) f is @f -pseudoconvex and @f -protoconvex at x 2 C; (b) f is @f -colinvex at x in the sense: there exists

: C Graph(@f ) ! P such that for all w 2 C; x 2 @f (x); f (w) f (x) + (w; x; x )hx ; w xi: (7) Proof . (b))(a) is obvious: given w 2 C; if f (w) < f (x) then hx ; w xi < 0 for all x 2 @f (x); while if f (w) f (x) then hx ; w xi 0 for all x 2 @f (x): Hence, f is @f -pseudoconvex and @f -protoconvex at x. (a))(b) Let f be @f -pseudoconvex and @f -protoconvex at x. Let w 2 C and x 2 @f (x):
If hx ; w xi = 0 then one has f (w) f (x) by @f -pseudoconvexity of f and one can take (w; x; x ) = 1 or any element in P:

If hx ; w xi > 0 then f (w) > f (x) by @f -protoconvexity of f . Then one can take (w;

x; x ) = f (w) f (x) hx ; w xi > 0:
If hx ; w xi < 0 then one can take (w;

x; x ) = maxf f (w) f (x) hx ; w xi ; 0g + 1, or any > 0 such that f (w) f (x)
hx ; w xi : For (w; u; u ) 2 C @f with u 6 = x we can take any value for (w; u; u ):

A similar result is valid for eventually colinvex functions i.e. functions such that for every w; x 2 C there exists x 2 @f (x) and (w; x; x ) 2 P such that

f (w) f (x) (w; x; x )hx ; w xi:
Proposition 2.2 For any function f : C ! R, x 2 domf and any @f : C X , the following assertions are equivalent.

(a) f is eventually @f -pseudoconvex and eventually @f -protoconvex at x; (b) f is eventually @f -colinvex at x:

3 Characterizations of generalized a¢ ne functions

In the present section we study classes of generalized a¢ ne functions which are still more restrictive. Their interests lie in their striking properties. In particular, they enjoy nice composition properties with the classes introduced in the preceding section. Their behaviors will be studied below.

First, we shall consider some composition properties.

Proposition 3.1 Let g : C ! R be (eventually) @g-colin…ne. Let ' : R ! R and f := ' g and let @f (x) := @'(g(x))@g(x):

(a) If ' is (eventually) @'-pseudoconvex, then f is (eventually) @f -pseudoconvex; (b) If ' is (eventually) @'-quasiconvex, then f is (eventually) @f -quasiconvex; (c) If ' is (eventually) @'-protoconvex, then f is (eventually) @f -protoconvex; (d) If ' is (eventually) @'-colin…ne, then f is (eventually) @f -colin…ne.
Proof. Since g is @g-colin…ne, for all w; x 2 C and all x 2 @g(x) there exists g (w; x; x ) 2 P such that g(w) g(x) = g (w; x; x )hx ; w x: Let u := '(x) and v := '(w):

(a) Since ' is @'-pseudoconvex, if there exists x f 2 @f (x) such that hx f ; w xi 0 then there exist x g 2 @g(x) and u 2 @'(u) such that x f = u x g 0: Moreover,

x f = u x g 0 ) u (g(w) g(x)) 0 ) '(g(w)) '(g(x)):
(b) Since ' is @'-quasiconvex, if there exists x f 2 @f (x) such that hx f ; w xi > 0 then there exist x g 2 @g(x) and u 2 @'(u) such that x f = u x g > 0: Moreover,

x f = u x g > 0 ) u (g(w) g(x)) > 0 ) '(g(w)) '(g(x)):
(c) A similar proof can be given when ' is @'-protoconvex.

(d) Now, if ' is @'-colin…ne, then for all x f 2 @f (x); there exist some x g 2 @g(x) and some u 2 @'(u) such that x f = u x g : Moreover, one has

'(g(w)) '(g(x)) = ' (v; u; u )hu ; g(w) g(x)i = ' (v; u; u ) g (w; x; x g )hu x g ; w xi hence f is @f -colin…ne.
When g is eventually @g-colin…ne, for w; x 2 C there exist x g 2 @g(x) and g (w; x; x ) 2 P such that g(w) g(x) = g (w; x; x )hx g ; w xi. Let u := '(x) and v := '(w):

Let ' be eventually @'-pseudoconvex. If '(g(w)) '(g(x)); then there exists u 2 @'(u) such that hu ; v ui 0: Hence '(g(w)) '(g(x)) ) hu ; g(w) g(x)i 0 ) hu x g ; w xi 0:

Clearly, u x g 2 @f (x): Thus, f is eventually @f -pseudoconvex.

Similar proofs can be given for (b), (c) and (d).

From the preceding proposition and De…nition 2.3, we have the following example show simple contructions of colin…ne functions.

Example 3.1 Let g; h : C ! R, g be @g-colin…ne and h be @h-colin…ne and let x 2 C:

(a) Let f := g + h or f := gh. If P@g = P@h and if @f P@g; then f is @f -colin…ne.

(b) Let f ( ) := 1 g( ) and D := fx 2 C : g(x) > 0g: If @f P@g, then f is @f -colin…ne on D:
Let us turn to properties of colin…ne functions. The following result is an easy consequence of Lemma 2.1 and Lemma 2.2. Proposition 3.2 (a) A function f is @f -colin…ne if, and only if, f is @f -colinvex and f is @f -colinvex.

(b) Suppose that for all x 2 C the set @f (x) is convex. If f is eventually @f -colinvex and f is eventually @f -colinvex, then f is eventually @f -colin…ne.

(c) Let C be open and let f be radially continuous. If f is (eventually) @f -protoconvex and f is (eventually) @f -protoconvex and 0 = 2 @f (x) for all x 2 C then f is (eventually) @f -colin…ne.

Proof. (a) Let f be @f -colin…ne. Then, there exists : C Graph(@f ) ! P such that for all w; x 2 C; x 2 @f (x); f (w) f (x) = (w; x; x )hx ; w xi: Thus, f is @f -colinvex and f is @f -colinvex since this function will satisfy relation [START_REF] Demyanov | Contructive nonsmooth analysis[END_REF] both for f and f . Conversely, let w; x 2 C. Since f is @f -colinvex and f is @f -colinvex, there exist 1 : C Graph(@f ) ! P and 2 : C Graph(@f ) ! P such that for all w; x 2 C; x 2 @f (x);

f (w) f (x) 1 (w; x; x )hx ; w xi; f (w) + f (x) 2 (w; x; x )h x ; w xi: Thus, if f (w) f (x) < 0 then hx ; w xi < 0, while if f (w) f (x) > 0 then hx ; w xi > 0; hence in both cases we can take (w; x; x ) = f (w) f (x) hx ; w xi > 0: Now when f (w) = f (x)
then hx ; w xi = 0 and one can take any (w; x; x ) 2 P. (b) If f is eventually @f -colin…ne, then as in the proof of (a), one gets that f is eventually @f -colinvex and f is eventually @f -colinvex.

Conversely, let w; x 2 C. Since f is eventually @f -colinvex and f is eventually @f -colinvex, there exist If f (w) f (x) < 0 then hg 1 (w; x); w xi < 0; hence we can take g(w; x) := g 1 (w; x), (w;

x) = f (w) f (x) hg 1 (w; x); w xi > 0; if f (w) f (x) > 0 then hg 2 (w;
x); w xi > 0 and we can take g(w; x) := g 2 (w; x), (w;

x) = f (w) f (x) hg 2 (w; x); w xi > 0; while if f (w) = f (x)
then hg 1 (w; x); w xi 0 and hg 2 (w; x); w xi 0 and, since @f (x) is convex, there exists g 3 (w; x) 2 X such that hg 3 (w; x); w xi = 0 and we can take g(w; x) := g 3 (w; x) and any (w; x; x ) 2 P.

(c) Under the assumptions of (c),

f is @f -pseudoconvex and f is @f -pseudoconvex (see [15, Proposition 2.5]). By Proposition 2.1, f is @f -colinvex and f is @f -colinvex. Hence from (a) f is @f -colin…ne.
Similarly, if f is eventually @f -protoconvex and f is eventually @f -protoconvex and 0 = 2 @f (x) for all x 2 C then f is (eventually) @f -colin…ne.

It may happen that f is eventually @f -colinvex and f is eventually @f -colinvex but f is not eventually @f -colin…ne when @f is not convex.

Example 3.2 Let f : R ! R be given by f (x) = 0 for x 0 and f (x) = x for x 0 and let @f be given by @f (x) = f 1; 1g for all x 2 C := X: Then f is eventually @f -colinvex and f is eventually @f -colinvex but f is not eventually @f -colin…ne since f ( 1) = f (0) = 0 and h1; 1 0i 6 = 0, h 1; 1 0i 6 = 0.

Proposition 3.3 (a) If f is @f -colin…ne on C, then f is quasia¢ ne (i.e. quasiconvex and quasiconcave) on C.

(b) If f is @f -colin…ne at x 2 C; then, setting @( f )(x) := @f (x), f is @f -protoa¢ ne and @f -pseudoa¢ ne at x: Proof. (a) Suppose that f is not quasia¢ ne: there exist x; y; z 2 C with y 2]x; z[such that either f (y) > maxff (x); f (z)g or f (y) < minff (x); f (z)g: If f (y) > maxff (x); f (z)g then for y 2 @f (y); one has hy ; x yi < 0 and hy ; z yi < 0; a contradiction with the linearity of y : If f (y) < minff (x); f (z)g a similar proof yields a contradiction.

(b) Since f is @f -colinvex and f is @f -colinvex, the assertion follows from Remark 1 and Proposition 2.1. Now, let us give a partial converse of the preceding result. It uses the following assumptions in which

x 2 C: 8u 2 T r (C; x) D + f (x; u) > 0; D + f (x; u) 0 ) D + f (x; u) > 0; (8) 8x 2 @f (x) 8u 2 T r (C; x) hx ; ui D + f (x; u) (9) 8x 2 @( f )(x) 8u 2 T r (C; x) hx ; ui D + ( f )(x; u) ( 10
)
8x 2 @f (x) 8u 2 T r (C; x) D + f (x; u) hx ; ui (11)
Obviously, condition ( 8) is satis…ed when f has a radial derivative at x in all directions u 2 T r (C; x): Moreover, if C is open, f is di¤erentiable on C and @f (x) = ff 0 (x)g; @( f )(x) = f f 0 (x)g; the other conditions are satis…ed. However, these conditions allow x to be a boundary point of C (which is not assumed to be open in the next results).

Let us recall that a function f is radially nonconstant if one cannot …nd any proper line segment on which f is constant. 9) is satis…ed, then f is @f -colinvex at x. In addition to [START_REF] Jeyakuma | On characterizing the solution sets of pseudolinear programs[END_REF], if one of the following two conditions holds, then f is @f -colin…ne at x.

Proposition 3.4 Let f be @f -pseudoa¢ ne on a convex set C. Then f is quasia¢ ne on C. Let x 2 C: (a) If f is radially nonconstant and if @( f ) = @f; then f is @f -colin…ne. (b) If relation (
(i) @( f )(x) = @f (x) and relation [START_REF] Komlosi | First and second characterizations of pseudolinear functions[END_REF] is satis…ed, (ii) conditions ( 8), ( 11) are satis…ed at x and for all u 2 V (C; x); there exists y 2 @( f )(x) such that D + ( f )(x; u) hy ; ui:

Proof. Since f is @f -pseudoconvex and f is @( f )-pseudoconvex, f and f are quasiconvex by [START_REF] Linh | Generalized convex functions and generalized di¤ erentials[END_REF]Proposition 6]. Thus, f is quasia¢ ne on C.

(a) Let us …rst prove that f is @f -protoconvex. Suppose that f is not @f -protoconvex. Then there exist some w; x 2 C such that f (w) f (x) and some x 2 @f (x) with hx ; w xi > 0: Then f (w) = f (x): On the other hand, since f is quasiconvex and radially nonconstant, there exists z 2 (w; x) such that f (z) < f (x) and then hx ; z xi < 0, a contradition.

Similarly, if f is not @( f )-protoconvex, then there exist some w; x 2 C such that f (w) f (x) and some x 2 @( f )(x) with hx ; w xi > 0: Since f is @( f )-pseudoconvex, one has f (w) = f (x): Since f is quasiconcave and radially nonconstant, there exists z 2 (w; x) such that f (z) > f (x), i.e. f (z) < f (x) and then hx ; z xi < 0, a contradition. Thus, f is @( f )-protoconvex.

On the other hand, since @( f ) = @f; f is @f -protoconvex and f is @f -pseudoconvex. Hence, f is @f -colinvex and f is @f -colinvex. By Proposition 3.2 (a), f is @f -colin…ne at x: (b) When relation ( 9) holds, we see that f is @f -protoconvex at x, since if f (w) f (x) for some w 2 C; then D + f (x; w x) 0 by quasiconvexity and for all x 2 @f (x) we have hx ; w xi D + f (x; w x) 0: Thus, since f is @f -pseudoconvex, it is @f -colinvex at x by Proposition 2.1.

Let us …rst consider the case (i): @( f )(x) = @f (x) and relation ( 10) is satis…ed. Since f is @f -pseudoa¢ ne, f is @( f )-pseudoconcave, hence by De…nition 2.2, f is @f -pseudoconvex. Now, let us prove that f is @( f )-protoconcave at x. If f (w) f (x) for some w 2 C, then for all x 2 @( f )(x) one has hx ; w xi D + ( f )(x; w x) 0 by relation [START_REF] Komlosi | First and second characterizations of pseudolinear functions[END_REF] and the fact that f is quasiconvex. Thus, f is @f -protoconvex at x and f is @f -colinvex at x. By Proposition 3.2 (a), f is @f -colin…ne at x:

Let us suppose assumption (ii) is satis…ed.

Let us prove that f is @f -protoconvex if relation ( 11) is satis…ed. If f (w) f (x) for some w 2 C; then f (w) f (x) and D + f (x; w x) 0 since f is quasia¢ ne. For any x 2 @f (x) relation ( 11) implies 0 D + f (x; w x) hx ; w xi. Thus one has h x ; w xi 0 and f is @f -protoconvex at x. Now, let us prove that f is @f -pseudoconvex. Since f is @f -pseudoa¢ ne, f is @( f )-pseudoconvex. Let w 2 C be such that f (w) < f (x) and let u := w x. Then we have hy ; w xi < 0 for all y 2 @( f )(x): Let us pick y 2 @( f )(x) such that D + ( f )(x; u) hy ; ui. Thus, we have D + f (x; w x) > 0. Since f is quasia¢ ne and f (w) > f (x), we also have D + f (x; w x) 0. Then condition (8) entails 0 < D + f (x; w x): Since relation ( 11) is satis…ed, for all x 2 @f (x) we have 0 < D + f (x; w x) hx ; w xi. Thus, we get h x ; w xi < 0 for all x 2 @f (x) and then f is @f -pseudoconvex.

Thus, f is @f -colinvex at x and then f is colin…ne at x: Let us give a similar result for an eventually colin…ne function at some x 2 C. It uses the following assumptions which are weaker than the preceding assumptions:

8u 2 T r (C; x) 9x 2 @f (x) hx ; ui D + f (x; u) (12) 8u 2 T r (C; x) 9x 2 @( f )(x) hx ; ui D + ( f )(x; u) ( 13
)
8u 2 T r (C; x) 9x 2 @f (x) D + f (x; u) hx ; ui (14)
Proposition 3.5 Let f be @f -pseudoa¢ ne on a convex set C and let x 2 C. (a) If f is radially nonconstant and if @( f ) \ @f 6 = ?; then f is eventually @f -colin…ne. (b) If relation to ( 12) is satis…ed, then f is eventually @f -colinvex. In addition, if @f (x) is convex and one of the following conditions holds, then f is eventually @f -colin…ne at x.

(i) @( f )(x) \ @f (x) 6 = ? and relation [START_REF] Komlosi | First and second characterizations of pseudolinear functions[END_REF] is satis…ed, (ii) @( f )(x) = @f (x) and relation ( 13) is satis…ed, (iii) conditions ( 8), ( 14) are satis…ed at x and for all u 2 T r (C; x) there exists y 2 @( f )(x) such that D + ( f )(x; u) hy ; ui: Proof. By the …rst assertion of the preceding proposition, f is quasia¢ ne on C: (a) As in the proof of the preceding proposition (a), f is @f -protoconvex and f is @( f )-protoconvex. Moreover, since @( f )\ @f 6 = ?; f is eventually @f -protoconvex and f is eventually @f -pseudoconvex. Hence, f is @f -colinvex and f is eventually @f -colinvex. By Proposition 3.2 (b), f is eventually @f -colin…ne at x: (b) Let us …rst observe that f is eventually @f -protoconvex at x, since if f (w) f (x) for some w 2 C; then D + f (x; w x) 0, and, by relation [START_REF] Kruk | Pseudolinear programming[END_REF], there exists x 2 @f (x) such that hx ; w xi D + f (x; w x) 0: Since f is also @f -pseudoconvex, it is eventualy @f -colinvex at x by Proposition 2.2.

In order to show that f is eventually @f -colin…ne at x; let us …rst consider the case (i). Since f is @f -pseudoa¢ ne, f is @( f )-pseudoconcave, hence by De…nition 2.2, f is @( f )-pseudoconvex. Since relation [START_REF] Komlosi | First and second characterizations of pseudolinear functions[END_REF] is satis…ed, as in the proof of the preceding proposition, we get that f is @( f )-protoconvex at x. Thus, f is @( f )-colinvex. On the other hand, since @( f )(x) \ @f (x) 6 = ?, f is eventually @f -pseudoconvex and eventually @f -protoconvex at x. Thus, f is eventually @f -colin…ne at x by Proposition 3.2 (b). Now, let us suppose assumption (ii) is satis…ed. First, we observe that f is eventually @( f )-protoconcave at x: since if f (w) f (x) for some w 2 C, then for some x 2 @( f )(x) one has hx ; w xi D + ( f )(x; w x) 0 by relation [START_REF] Linh | Optimality conditions for quasiconvex programming[END_REF]. On the other hand, since @( f )(x) = @f (x), f is eventually @f -protoconvex at x. So, f is eventually @f -colin…ne at x:

Finally, let us suppose assumption (iii) is satis…ed. Let us …rst prove that f is eventually @f -protoconvex at x. If f (w) f (x) for some w 2 C; then f (w) f (x) and D + f (x; w x) 0 since f is quasia¢ ne. By relation [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF], there exists x 2 @f (x) such that 0 D + f (x; w x) hx ; w xi. Thus one has x 2 @f (x) with h x ; w xi 0: f is eventually @f -protoconvex at x. Now, let us prove that f is eventually @f -pseudoconvex at x. Since f is @f -pseudoa¢ ne, f is @( f )pseudoconvex. If f (w) < f (x) for some w 2 C; using our assumption and condition [START_REF] Jeyakuma | Nonsmooth vector functions and continuous optimization[END_REF] as in the preceding proof, we get 0 < D + f (x; w x): Since relation ( 14) is satis…ed, there exists x 2 @f (x) such that D + f (x; w x) hx ; w xi. Thus, we get x 2 @f (x) with h x ; w xi < 0: f is eventually @f -pseudoconvex at x.

Thus, f is eventually @f -colinvex at x and then f is eventually @f -colin…ne at x: Remark 3.1 (a) Note that if we change " " into "<" in relations ( 11), ( 9), then assumption [START_REF] Jeyakuma | Nonsmooth vector functions and continuous optimization[END_REF] in the preceding results could be deleted. (b) Clearly, condition ( 8) is satis…ed if f has a radial derivative in the direction u for all u 2 T r (C; x): (c) The preceding proposition implies [14, Proposition 13].

Proposition 3.6 Let f be @f -colin…ne on C and let w; x 2 C: For r 2 R, let x r := x + r(w x): Then for any s; t 2 R; with s 6 = t such that x s ; x t 2 C; the following statements are equivalent:

(a) f (x) = f (w); (b) f (x t ) = f (x s ); (c) hx t ; x s x t i = 0 for all x t 2 @f (x t ); (d) hx s ; x t x s i = 0 for all x s 2 @f (x s ).
Proof. (a))(b) If f (x) = f (w) then for any x 2 @f (x), hx ; w xi = 0 and then, 0 = hx

; x t xi = hx ; x s xi. By De…nition 2.3, f (x t ) = f (x) = f (x s ).
(b),(c),(d) is obvious by De…nition 2.3. (c))(a) If hx t ; x s x t i = 0 for all x t 2 @f (x t ) then, since s 6 = t; for all x t 2 @f (x t ) one has hx t ; x wi = 0, hence hx t ; x x t i = hx t ; w x t i = 0 and then f

(x) = f (x t ) = f (w).
For an eventually colin…ne function, we have the following property in the spirit of Minty variational inequalities and Stampachia variational inequalities. It shows that, in our case, a solution in the sense of Minty is a solution in the sense of Stampachia.

Recall ( [START_REF] Aussel | On quasivariational inequalities[END_REF]) that a multimap T : C X is said to be upper sign-continuous on C if, for every w; x 2 C; the following implication (in which x t := tx + (1 t)w) holds: 8t 2 (0; 1); inffhx t ; w xi : x t 2 T (x t )g 0 ) supfhx ; w xi : x 2 T (x)g 0: Proposition 3.7 Let f be eventually @f -colin…ne on C and let w; x 2 C:

(a) If for all w 2 @f (w) one has hw ; w xi 0 (resp. hw ; w xi > 0, hw ; w xi = 0, hw ; w xi < 0, hw ; w xi 0) then there exists x 2 @f (x) such that hx ; w xi 0 (resp. hx ; w xi > 0, hx ; w xi = 0, hx ; w xi < 0, hx ; w xi 0).

(b) @f is upper sign-continuous on C:

Proof. (a) Since f is eventually @f -colin…ne on C; there exist w 2 @f (w); x 2 @f (x), (w; x; x ) 2 P and (x; w; w ) 2 P such that f (w) f (x) = (w; x; x )hx ; w xi = (x; w; w )hw ; x wi:

If for all w 2 @f (w) one has hw ; w xi 0; then the preceding relation entails f (w) f (x). Thus, hx ; w xi 0: The proofs of the other assertions are similar.

(b) Since x t x = t(w x) for t 2 (0; 1); (b) follows from (a) by substituting x t ; x t to w; w :

The preceding proposition can easily adapted to the case f is @f -colin…ne by interchanging "for all" and "there exists". Moreover, in such a case, one sees @f and @f are pseudomonotone (in the sense of [START_REF] Penot | Generalized convexity of functions and generalized monotonicity of set-valued maps[END_REF]; see also [START_REF] Penot | Are generalized derivatives useful for generalized convex function[END_REF], [START_REF] Penot | Generalized monotonicity of subdi¤ erentials and generalized convexity[END_REF]). Thus we get a multivalued generalization of the concept of PPM map which has been used in [START_REF] Bianchi | An extension of pseudolinear functions and variational inequalities problems[END_REF] to study variational inequalities.

The following result is similar to [14, Proposition 10].

Proposition 3.8 If f is @f -colin…ne then the function t 7 ! f (x + t(w x)) is either increasing or decreasing or constant on the interval on which it is de…ned.

Proof. Since f is @f -colin…ne, there exists : C Graph(@f ) ! P such that for every w; x 2 C; for any x 2 @f (x); f (w) f (x) = (w; x; x )hx ; w xi:

If f (w) = f (x)
then by Proposition 3.6, one has f (x + t(w x)) = f (x) for all t 2 R such that x t := x + t(w x) 2 C: If f (w) > f (x) then for any x 2 @f (x) one has hx ; w xi > 0: For all t > 0 such that x t := x+t(w x) 2 C; by homogeneity, one gets hx ; x t xi > 0. Thus f (x t ) > f (x): Also, given u := x + r(w x) 2 C and v := x + s(w x) 2 C with s > r; one has f (v) > f (u): Otherwise, one would have either f (v) = f (u) and then f (w) = f (x) by what precedes, or f (v) < f (u) and then, for all v 2 @f (v), hv ; u vi > 0; hence hv ; x vi > 0 since x v = q(u v) for some q > 0 and then f (x) > f (v); a contradiction with f (v) = f (x s ) > f (x). A similar proof shows that for every t > 0 we have f (w + t(x w)) < f (w) and for s > r; f (w + s(x w)) < f (w + r(x w)) when the involved points are in C:

If f (x) > f (w); then we also have the conclusion by interchanging the roles of w and x in what precedes.

In the sequel, we will consider the following hypothesis: (H + ) If hx ; w xi > 0 for some w; x 2 C and some x 2 @f (x); then there exists z 2]w; x[ such that f (z) > f (x).

(H ) If hx ; w xi < 0 for some w; x 2 C and some x 2 @f (x); then there exists z 2]w; x[ such that f (z) < f (x).

(H 0 ) If hx ; w xi = 0 for some w; x 2 C and some x 2 @f (x); then one has f (w) = f (x).

Hypothesis (H + ) (resp. (H )) is weaker than @f -protoconvexity of f (resp. @f -protoconvexity of f ). Hypothesis (H + ) (resp. (H )) is clearly satis…ed when @f (x) (resp. @f (x)) is contained in the Dini-Hadamard (or contingent) subdi¤erential (resp. @ D ( f )(x) of f at x).

The following theorem has been given in [START_REF] Cambini | Generalized convexity and optimality conditions in scalar and vector optimization[END_REF]Theorems 4.13,4.14], [START_REF] Chew | Pseudolinearity and e¢ ciency[END_REF], [START_REF] Komlosi | First and second characterizations of pseudolinear functions[END_REF], [START_REF] Kortanek | Pseudoconcave programming and Lagrange regularity[END_REF], in the case the function f is di¤erentiable and @f := ff 0 g: Here, the function f is nonsmooth. In [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF], a generalized directional derivative is used instead of a generalized di¤erential. We start with a preparatory lemma dealing with the one-dimensional case.

Lemma 3.1 Let C be an open interval of R and let ' : C ! R be a continuous function satisfying (H + ) and (H ) with respect to @'. Then ' is @'-colin…ne if, and only if, either (H 0 ) is satis…ed and there exists some t 0 2 C such that @'(t 0 ) 3 f0g; or for all t 2 C the sign of t 2 @'(t) is constant on the level set L ' (t) := ' 1 ('(t)).

Proof. Let ' be @'-colin…ne. If there exists t 0 2 C such that 0 2 @'(t 0 ); then by the proof of the preceding proposition, f is constant on C: Thus, (H 0 ) is satis…ed and, since ' is @'-colin…ne, one has t = 0 for any t 2 @'(t) and any t 2 C:

Otherwise, for all t 2 C; t 2 @'(t); one has t 6 = 0: Thus, by Proposition 3.8, ' is increasing or decreasing on C and then the level set L ' (t) of ' at t 2 C is the singleton ftg: Now, let t 1 , t 2 2 @'(t) and let s 2 C; s > t: Since ' is @'-colin…ne there exist 1 (s; t; t 1 ) 2 P and 2 (s; t; t 2 ) 2 P such that '(s) '(t) = 1 (s; t; t 1 )ht 1 ; s ti = 2 (s; t; t 2 )ht 2 ; s ti:

Thus t 1 t 2 > 0.
For the converse, let us …rst suppose (H 0 ) is satis…ed and for some t 0 2 C we have @'(t 0 ) 3 f0g. Then, for all s 2 C; 0 = t 0 2 @'(t 0 ); we have ht 0 ; s t 0 i = 0; so that, by (H 0 ), '(s) = '(t 0 ): Hence, ' is constant and, in view of conditions (H + ) and (H ), for all r 2 C; r 2 @'(r); we must have r = 0: then ' is trivially @'-colin…ne. Now suppose that for all t 2 C, t 2 @'(t), the sign of t is constant on L ' (t). Given r 2 C such that for all r 2 @'(r); r > 0; we will prove that '(t) > '(r) for all t > r: Let us …rst prove that we cannot have '(t) < '(r): If '(t) < '(r), let s := supfp 2 [r; t] : '(p) = '(r)g: Since ' is continuous, we have s < t and '(s) = '(r): Then our assumption ensures that, for all s 2 @'(s); we have s > 0;so that, by condition (H + ) there exists some p 2]s; t[ with '(p) > '(s) and the intermediate value theorem yields some q 2 [p; t] with '(q) = '(s) = '(r); a contradiction with the de…nition of s: Now let us suppose that '(t) = '(r): Then t > 0 for any t 2 @'(t) by our assumption, so that ht ; r ti < 0: Then condition (H ) yields some t 0 2]r; t[ such that '(t 0 ) < '(t) = '(r) and replacing t by t 0 in what precedes, we get again a contradiction.

A similar proof shows that '(q) < '(r) for all q < r:

We can see in a similar way that when for some r 2 C; and all r 2 @'(r) we have r < 0, the function ' is decreasing on C:

Hence, for any r; s 2 C; s 2 @'(s) there exists some (r; s; s ) 2 P such that '(r) '(s) = (r; s; s )hs ; r si:

We are ready to prove our main result. It is a nonsmooth version of [5, Theorems 4.13, 4.14], [START_REF] Chew | Pseudolinearity and e¢ ciency[END_REF], [START_REF] Komlosi | First and second characterizations of pseudolinear functions[END_REF], [START_REF] Kortanek | Pseudoconcave programming and Lagrange regularity[END_REF], [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF]. (a) Let f be @f -colin…ne. Then, for all w 2 C; the set K w := fu 2 X : 8w 2 @f (w); hw ; ui = 0g is a linear subspace of X and @f (w) is contained in an half line.

(b) If f is @f -colin…ne on C then for any w; x 2 C such that f (w) = f (x) one has K x = K w and for any w 2 @f (w); x 2 @f (x); there exists some r > 0 such that w = rx :

(c) Conversely, suppose that f and @f satisfy (H 0 ), (H + ) and (H ). If for any w; x 2 C such that f (w) = f (x) and for any w 2 @f (w); x 2 @f (x), there exists some r > 0 such that w = rx ; then f is @f -colin…ne.

(d) Assume that the dimension of X is greater than 1; that C = X, and that f and @f satisfy (H 0 ), (H + ) and (H ). Then f is @f -colin…ne if, and only if, for all w; x 2 C and all w 2 @f (w); x 2 @f (x) there exists some r > 0 such that w = rx .

Proof. (a) It is obvious that K w is a linear subspace. Let w ; x 2 @f (w): For any u 2 X such that hw ; ui = 0; taking t > 0 small enough, we have w + tu 2 C and f (w + tu) = f (w): Again, since f is @f -colin…ne, we obtain hx ; ui = 0: Thus, ker w = K w and x jjx jj = w jjw jj or x jjx jj = w jjw jj if w and x are both non null, while if one is 0; the other one is 0 too. In other terms, there exists some r 2 Rnf0g such that w = rx : Let us prove that r is positive when x 6 = 0. Let y 2 C be such that hx ; y xi 6 = 0: Since f is @f -colin…ne, one has (b) Now let w; x 2 C be such that f (w) = f (x) and let u 2 K w . Since f is @f -colin…ne, we have x w 2 K w , hence x w+tu 2 K w for all t 2 R since K w is a linear subspace. Thus, for jtj small enough, we have x+tu 2 C and f (x + tu) = f (w) = f (x): Therefore hx ; ui = 0 for all x 2 @f (x) and u 2 K x . So, K w K x . The symmetry of the roles of w and x yields K w = K x .

Let w 2 @f (w); x 2 @f (x): By (a) and the preceding case, we have ker w = K w = K x = ker x and so there exists some r 2 Rnf0g such that w = rx : Let us prove that r is positive. Suppose that r < 0. Pick u 2 X such that hx ; ui = 1 and set z 1 = w + tu and z 2 = x + tu, with t > 0 small enough to ensure that z 1 ; z 2 2 C: Then, since f is @f -colin…ne, one has f (z 1 ) < f (w) = f (x) < f (z 2 ) and hx ; w xi = 0: Since f is radially continuous, there exists some s 2]0; 1[ such that, for z := sz 1 + (1 s)z 2 ; one has f (z) = f (x): Then z x = sw sx + tu and 0 = hx ; z xi = hx ; sw sx + tui = t > 0; a contradiction. Hence, the case r < 0 is excluded. (c) If there exist x 2 C such that @f (x) 3 0; then one has f (y) = f (x) for all y 2 C by condition (H 0 ) since hx ; y xi = 0: Thus, the level set of f at x is C and our assumption ensures that for all y 2 @f (y); for some r > 0; we have y = rx = 0. Thus f is @f -colin…ne.

(b) there exist a continuous linear form `on X; `6 = 0 and a lower semicontinuous (resp. continuous) increasing, @g-colin…ne function g : R ! R such that f = g `:

Proof. (a)=)(b) Let condition (C1) be satisfy and let f : X ! R be l.s.c. (resp. continuous) and @f -colin…ne. Hence f is quasia¢ ne by Proposition 3.3. According to [START_REF] Martinez-Legaz | Quasiconvex duality theory by generalized conjugation methods[END_REF], [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF], there exist a continuous linear form `on X and a lower semicontinuous. (resp. continuous) nondecreasing function g : R ! R such that f = g `: Since f is nonconstant, `6 = 0: Let us prove that g is increasing. Suppose that there exist r 1 < r 2 in R such that for all r 2 [r 1 ; r 2 ]; g(r) = g(r 1 ) = g(r 2 ). Since f is nonconstant, g is nonconstant and there exists some t 0 > 1, such that, for t t 0 one has either g(r 1 + t(r 2 r 1 )) > g(r 2 ) or g(r 2 + t(r 1 r 2 )) < g(r 1 ): We consider the …rst case, the second one being similar. Since `6 = 0, there exist w; x 2 X such that `(w) = r 1 < `(x) = r 2 . Then, for t t 0 ; we have `(w + t(x w)) > `(x) = r 2 and then f (w + t(x w)) = g(`(w + t(x w))) = g(r 1 + t(r 2 r 1 )) > g(r 2 ) = f (w) = f (x): By Proposition 3.8, f is not @f -colin…ne. Therefore g is increasing.

Now, we shall prove that g is @g-colin…ne. For any u; v 2 R, and v 2 @g(v); by condition (C1), there exists w; x 2 X and x 2 @f (x) such that `(w) = u, `(x) = v and v `= x : Hence there exists f (w; x; x ) 2 P such that g(u) g(v)

f (w; x; x ) = g(`(w)) g(`(x))

f (w; x; x ) = hx ; w xi = hv ; `(w) `(x)i = hv ; u vi: so that g is @g-colin…ne.

(b))(a) Let condition (C2) be satisfy and w; x 2 X. Then, for any x 2 @f (x); there exists u 2 @g(`(x)) such that x = u `: Since g is @g-colin…ne, there exists g (`(w); `(x); u ) 2 P such that f (w) f (x) g (`(w); `(x); u ) = g(`(w)) g(`(x))

g (`(w); `(x); u ) = hu ; `(w) `(x)i = hu `; w xi = hx ; w xi hence, f is @f -colin…ne.

Characterizations of solution sets

In the present section, we consider the minimization problem

(C) min f (x) subject to x 2 C
where C is a subset of n.v.s. X and f : C ! R.

Our purpose here is limited: since optimality conditions for (C) and mathematical programming problems using the concepts of the previous sections are dealt with in [START_REF] Linh | Optimality conditions for quasiconvex programming[END_REF], [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF], [START_REF] Linh | Generalized convex functions and generalized di¤ erentials[END_REF], [START_REF] Penot | Are generalized derivatives useful for generalized convex function[END_REF], [START_REF] Penot | Characterization of solution sets of quasiconvex programs[END_REF], [START_REF] Penot | A Lagrangian approach to quasiconvex analysis[END_REF], we are just concerned with characterizations of solution sets. Let S be the set of solutions to (C) and let @f : C X be a generalized di¤erential of f:

The following results are direct consequences of De…nition 2.3. Here we set S a := fx 2 C : 9a 2 @f (a); ha ; x ai = 0g; S 0 a := fx 2 C : 8a 2 @f (a); ha ; x ai = 0g: Proposition 4.1 Let f be @f -colin…ne on C: (a) If a 2 S, then for all x 2 C; a 2 @f (a); x 2 @f (x) one has ha ; x ai 0 and hx ; a xi 0. Moreover, S = S a = S 0 a : (b) Conversely, if for all x 2 C; either there exists a 2 @f (a) such that ha ; x ai 0 or there exists x 2 @f (x) such that hx ; a xi 0, then a 2 S. Proposition 4.2 Let f be an eventually @f -colin…ne function on C:

(a) If a 2 S, then for all x 2 C, there exists some a 2 @f (a) such that ha ; x ai 0 and there exists some x 2 @f (x) such that hx ; a xi 0.

Moreover, S 0 a S S a : (b) Conversely, if for all x 2 C, either for all a 2 @f (a) one has ha ; x ai 0 or for all x 2 @f (x) one has hx ; a xi 0 then a 2 S.

1 :

 1 C C ! P, 2 : C C ! P and g 1 : C C ! X ; g 2 : C C ! X such that for all w; x 2 C; 2 (w; x)hg 2 (w; x); w xi f (w) f (x) 1 (w; x)hg 1 (w; x); w xi:

Theorem 3 . 1

 31 Let C be an open, convex subset of X; let f : C ! R be radially continuous and let @f : C X with nonempty values.

f

  (y) f (x) = (y; x; x )hx ; y xi = (y; x; w )hw ; y xi; hence we get 0 hx ; y xihw ; y xi = hx ; y xihrx ; y xi = rjjhx ; y xijj 2 and r > 0:

Thus, we can assume that 0 = 2 @f (x) for all x 2 C: Now, given w; x 2 C, for t 2 C w;x := ft 2 R : x t 2 Cg; where x t := (1 t)x + tw; let us set '(t) := f (x t ). For t 2 C w;x ; let us take @'(t) := ft 2 R : 9x t 2 @f (x t ); t = hx t ; w xig: Let us show that ('; @') satis…es (H + ), (H ) and (H 0 ). Suppose that there exists t 2 @'(t) such that ht ; s ti > 0 for t; s 2 C w;x , with s 6 = t: Then there exists x t 2 @f (x t ) such that hx t ; x s x t i = (s t)hx t ; w xi = (s t)t > 0: Since f satis…es (H + ), there exists q between t and s such that '(q) := f (x q ) > f (x t ) := '(t): Thus, ' satis…es (H + ). With a similar proof, we can see that ' satis…es (H ). Now suppose that ht ; s ti = 0 for t; s 2 C w;x , with s 6 = t and t 2 @'(t): Then there exists x t 2 @f (x t ) such that hx t ; x s x t i = (s t) t = 0:

Suppose that for all t 2 C w;x 0 = 2 @'(t). Let t; s 2 C w;x ; t 2 @'(t); s 2 @'(s) be such that '(t) = '(s). We shall prove that t s > 0: Let x t 2 @f (x t ) and x s 2 @f (x s ) be such that t = hx t ; w xi and s = hx s ; w xi: Since f (x t ) = '(t) = '(s) = f (x s ), there exists some r > 0 such that x s = rx t and we have s = hx s ; w xi = rhx t ; w xi = rt : Hence, we get t s > 0:

Thus, either there exists some t 2 C w;x such that 0 2 @'(t) or for all t 2 C the sign of t 2 @'(t) is constant on the level set L ' (t) := ' 1 ('(t)); hence ' is @'-colin…ne, by the preceding lemma.

Moreover, for any w; x 2 C, for any x 2 @f (x), we have s := hx ; w xi 2 @'(0) and

The su¢ ciency is clear by (c). Conversely, suppose that f is @f -colin…ne. Given w; x 2 C; w 2 @f (w) and x 2 @f (x) let prove there exists some r > 0 such that w = rx . If x = 0; since f is @f -colin…ne, then f is constant and so w = 0: thus the conclusion holds with r = 1 in this case or in the case w = 0: The conclusion also holds when f (w) = f (x); by (b). When w 6 = 0; x 6 = 0, f (w) 6 = f (x) and there is no r 2 Rnf0g such that w = rx the subspaces K w and K x are the hyperplanes ker w and ker x respectively, so that, the dimension of X being greater than 1; there exists z 2 (w + K w ) \ (x + K x ). One has z w 2 K w , z x 2 K x , hence hw ; z wi = 0 and hx ; z xi = 0. Since f is @f -colin…ne, we have f (x) = f (z) = f (w), a contradiction. Then, the existence of some r > 0 satisfying w = rx has been established in (b). Thus this case does not occur. It follows that when w 6 = 0 and x 6 = 0 there is some r 2 Rnf0g such that w = rx : Then we have for some ; > 0; f (x) f (w) = hw ; x wi = rhx ; x wi and f (w) f (x) = hx ; w xi; so that r = . Thus r is positive.

Note that [14, Lemma 17, Theorem 18] follow from the preceding lemma and the preceding theorem by taking @f (x) := f`(x)g. When @f (x) := frf (x)g; [5, Theorem 4.13, 4.14] is a consequence of the preceding theorem.

We devote the rest of this section, to a study of the structure of continuous colin…ne functions on a …nite dimensional n.v.s. X: When X is one-dimensional, it is easy to see that f is quasia¢ ne if, and only if, it is either nondecreasing or nonincreasing. Thus we suppose that the dimension of X is greater than 1: The di¤erentiable case is considered in [START_REF] Cambini | On generalized linearity of quadratic fractional functions[END_REF][START_REF] Cambini | Generalized convexity and optimality conditions in scalar and vector optimization[END_REF][START_REF] Komlosi | First and second characterizations of pseudolinear functions[END_REF][START_REF] Kortanek | Pseudoconcave programming and Lagrange regularity[END_REF] as a Df -colin…ne function f is pseudoa¢ ne (or pseudolinear in the terminology of these papers). The nonsmooth case is dealt with in [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF] with the help of a bifunction. Here, we use a generalized di¤erential @f of f and we assume that some natural composition rules are satis…ed.

We suppose that when f = g `for some non null continuous linear form `on X and some continuous function g : R ! R, one the following conditions is satis…ed:

(C1) @g(`(x)) ` @ (g `) (x); (C2) @ (g `) (x) @g(`(x)) `: Conditions (C1) and (C2) are satis…ed for the Fréchet and the Hadamard subdi¤erentials.

We deduce from [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF], [START_REF] Martinez-Legaz | Quasiconvex duality theory by generalized conjugation methods[END_REF], [START_REF] Penot | Glimpses upon quasiconvex analysis[END_REF] a characterization of continuous @f -colin…ne functions.

Proposition 3.9 Let f : X ! R be a lower semicontinuous (resp. continuous), nonconstant function on a …nite dimensional n.v.s. If condition (C1) is satis…ed, then assertion (a) below implies assertion (b); if condition (C2) is satis…ed, the reverse implication holds: (a) f is @f -colin…ne;

Let us consider now the case in which the constraint set C is de…ned by a …nite family of inequalities, so that problem (C) turns into the mathematical programming problem (M) min f (x) subject to x 2 C := fx 2 X : g 1 (x) 0; :::; g n (x) 0g;

where f : W ! R, g i : W ! R and W is a subset of X. Let a 2 C, let I := f1; :::; ng and let I(a) := fi 2 I : g i (a) = 0g:

In convex mathematical programming, the multipliers are the same for all solutions. The following result follows from [START_REF] Linh | Generalized a¢ ne maps and generalized convex functions[END_REF]. Note here instead of colinvexity of the objective function f; we only need that f is @fprotoconvex.

Proposition 4.3 Let a 2 S be such that there exists y = (y i ) 2 R n + ; a 2 @f (a), a i 2 @g i (a) for i 2 I(a) satisfying a + P i2I(a)

y i a i = 0 and y i g i (a) = 0 for i 2 I.

(a) If either f is @f -colin…ne at a or f is @f -protoconvex at a and g i is @g i -protoconvex at a (i 2 I(a)), then P i2I(a) y i ha i ; b ai = 0 for every b 2 S.

(b) Let f be @f -protoconvex at a and g i be @g i -colinvex at a (i 2 I(a)). Let L be the Lagrangian given by L(x; u) := f (x) + ug(x) for Now, assume that f is @f -protoconvex at a and g i is @g i -protoconvex at a