Zihao Guo

Jianqiang Zhou

Bruno Guelorget

Zhidan Sun

Delphine Retraint

Benoît Panicaud

EFFECT OF RESIDUAL STRESS ON THE TENSILE LOADING OF A GRADIENT MICROSTRUCTURE MATERIAL OBTAINED BY SURFACE MECHANICAL ATTRITION TREATMENT

Keywords: Gradient microstructure material, Residual stress, Surface mechanical attrition treatment, Tensile loading, Finite element modeling _________________________________________________________________________________________________________________________________

During the surface mechanical attrition treatment (SMAT), residual stress, work hardening and grain nano-crystallization are produced simultaneously, which allows to enhance the mechanical properties of materials. In this work, a dislocation density-based constitutive model is adapted to describe the local mechanical behavior of the gradient microstructure generated by SMAT and predict the macroscopic behavior of a SMATed cylindrical structure under tensile loading. A finite element (FE) model is built to perform uniaxial tensile simulation with the introduction of a residual stress field and a work hardening gradient obtained by X-ray diffraction (XRD). The results of simulation are in good agreement with the experimental results. Furthermore, it has been found that the residual stress plays a significant role in the initial stage of deformation. It decreases the yield stress of such gradient microstructure materials.

Introduction

Surface Mechanical Attrition Treatment (SMAT) is based on multi-directional impacts on the surface of a material with shot boosted by an ultrasonic generator [START_REF] Chen | Tensile properties of a nanocrystalline 316L austenitic stainless steel[END_REF][START_REF] Lu | Nanostructured surface layer on metallic materials induced by surface mechanical attrition treatment[END_REF]. The severe plastic deformation due to the remarkably increased strain and strain rate can produce a great number of microscopic defects, which progressively induces nano-crystallization in the superficial layer of materials. However, SMAT cannot affect the metallurgical features and mechanical properties in the bulk material. Hence, a gradient microstructure is generated from the treated surface to the bulk with continuously increased grain sizes [START_REF] Huang | Fatigue behaviors of AISI 316L stainless steel with a gradient nanostructured surface layer[END_REF][START_REF] Lei | Enhanced mechanical properties and corrosion resistance of 316L stainless steel by pre-forming a gradient nanostructured surface layer and annealing[END_REF][START_REF] Liu | Surface nanocrystallization of 316L stainless steel induced by ultrasonic shot peening[END_REF].

In the literature, various techniques such as Electron backscatter diffraction (EBSD), nanoindentation and X-ray diffraction (XRD), are widely used to investigate the morphological features and properties of gradient microstructure materials [START_REF] Fang | Revealing Extraordinary Intrinsic Tensile Plasticity in Gradient Nano-Grained Copper[END_REF][START_REF] Wu | Characterization of gradient properties generated by SMAT for a biomedical grade 316L stainless steel[END_REF][START_REF] Yin | Strain rate sensitivity of the ultrastrong gradient nanocrystalline 316L stainless steel and its rate-dependent modeling at nanoscale[END_REF]. However, it is difficult for experimental investigations to precisely characterize the mechanical properties at different depths, especially for the nanocrystallized layer. Therefore, in the literature, theoretical modeling is implemented to investigate the microscopic mechanical performances. For instance, a mechanism-based strain gradient plasticity model was modified to describe the mechanical response of gradient microstructure materials [START_REF] Li | Modeling of the plastic deformation of nanostructured materials with grain size gradient[END_REF][START_REF] Zhao | Multiple mechanism based constitutive modeling of gradient nanograined material[END_REF][START_REF] Zhu | Microstructures-based constitutive analysis for mechanical properties of gradient-nanostructured 304 stainless steels[END_REF]. In another work, the Kocks-Mecking-Estrin (KME) model was employed to describe the evolution of dislocation density with plastic strain rate and gradient grain size [START_REF] Kocks | Physics and phenomenology of strain hardening: the FCC case[END_REF].

Besides the gradient microstructure, high compressive residual stresses are inevitably induced in the near-surface region. The impacts also generate a gradient work hardening region, which could also have a significant effect on the mechanical behavior of materials [START_REF] Huang | Fatigue behaviors of AISI 316L stainless steel with a gradient nanostructured surface layer[END_REF]. In most work, the generated residual stresses are not considered in modeling, and only the work hardening gradient is taken into account through a dislocation density distribution [START_REF] Chen | Investigation of ductile damage during surface mechanical attrition treatment for TWIP steels using a dislocation density based viscoplasticity and damage models[END_REF]. In order to numerically and accurately investigate the mechanical properties of the gradient microstructure generated by SMAT, the residual stress and work hardening should be considered in the modelling.

This work aims thus to study the effect of residual stress on the properties of the gradient microstructure. For this purpose, the gradient microstructure of a SMATed 316L stainless steel is considered using a dislocation density and grain size based constitutive model. A 2Daxisymmetric cylindrical structure is built to analyze the uniaxial tensile mechanical behavior. Both the equilibrated residual stresses and the initial dislocation density distribution are implemented in the FE structure through user-defined subroutines.

Material and SMAT

Gradient microstructure obtained by SMAT

In this work, a dumbbell shape specimen of 316L stainless steel is treated on the entire gauge area, and the SMAT conditions are presented in a previous work [START_REF] Wu | Characterization of gradient properties generated by SMAT for a biomedical grade 316L stainless steel[END_REF]. As illustrated in Fig. 1, a gradient microstructure can be observed on the cross-section of a SMATed specimen through EBSD. The microstructure manifests a clear gradient feature from a nanocrystallized layer at the top treated surface to a coarse-grained microstructure in the bulk region. Due to the multidirectional severe plastic deformation induced by the impacts of flying balls, the grain size is refined and it ranges from 50 nm to 300 nm within a superficial nanocrystallized layer with a thickness of 5 μm.

Measurement of residual stress and work hardening

After the specimen was treated by SMAT, residual stresses and work hardening were measured by XRD. Work hardening cannot be directly measured, and it is usually evaluated by microscopic features (such as plastic slip and dislocation density). Iterative electrolytic polishing was performed to obtain their in-depth distributions in the near-surface region of the SMATed specimen [START_REF] Zhou | Reconstruction of residual stress and work hardening and their effects on the mechanical behaviour of a shot peened structure[END_REF]. The data points shown in Fig. 2a present the in-depth distribution of two residual stress components, respectively in the axial and circumferential directions. It can be seen that these two residual stresses exhibit compressive nature with a depth until about 500 µm. These residual stresses profiles will be taken into account in the further modeling.

As highlighted in Fig. 1, a large near-surface region is mechanically affected by SMAT due to high strain rate impacts. During the treatment, plastic strain is gradually accumulated in the near-surface region. In previous work [START_REF] Zhou | Reconstruction of residual stress and work hardening and their effects on the mechanical behaviour of a shot peened structure[END_REF], the residual accumulated plastic strain p rs  , which can describe the extent of work hardening, was assessed from FWHM of XRD peaks by Eq. ( 1). The obtained in-depth distribution of ΔFWHM is shown in Fig. 2b.

p rs FWHM     (1) max max 2.35( 0.68) 0, 0.68 r r FWHM where r r            (2)
where r and rmax are the distance to the axis of the cylindrical structure and the radius of the specimen, respectively;  is a coefficient of proportionality; FWHM  is the variation of FWHM between SMATed and untreated samples.

An equilibrium method for reconstruction of the residual stress field and the work hardening gradient is used in a cylindrical structure by FE modelling [START_REF] Zhou | Reconstruction of residual stress and work hardening and their effects on the mechanical behaviour of a shot peened structure[END_REF]. According to the elastic-plastic equilibrium and boundary conditions, the residual stresses and work hardening are recalculated to obtain the true residual stress field inside a cylindrical structure. a b Fig. 2. In-depth distribution of (a) residual stress, and (b) ΔFWHM whose fitted curve based on Eq. ( 2) is given.

Modelling of gradient microstructure materials

Constitutive model

In the framework of elastic-viscoplastic model, the total strain rate tensor ij  can be decomposed into elastic and plastic parts:

e p ij ij ij         (3)
The elastic part obeys the linear relationship between elastic strain rate and stress rate defined by Hooke's model:

1 2 9 e kk ij ij ij K            (4)
where 1 3

ij ij kk ij           is the deviatoric stress rate, kk
 is the hydrostatic stress rate, ij  is the Kronecker's symbol;  and K are the shear modulus and bulk modulus, respectively. The plastic strain rate tensor is related to the deviatoric stress 1 3

ij ij kk ij        based on the J2- flow theory of plasticity: 3 2 p p ij ij e         (5)
where 2 3

p p p ij ij        and 3 2 e i j i j      
are the equivalent plastic strain rate and the von Mises effective stress, respectively. In order to calculate without employing higher order stresses, a power law viscoplastic model [START_REF] Li | Modeling of the plastic deformation of nanostructured materials with grain size gradient[END_REF] is used to determine the effective plastic strain rate: where m (larger than 20) is the rate-sensitivity exponent; 0  is a reference strain rate; flow  is the flow stress. According to Taylor's hardening model and Hall-Petch strengthening [START_REF] Kashyap | On the Hall-Petch relationship and substructural evolution in type 316L stainless steel[END_REF][START_REF] Taylor | The mechanism of plastic deformation of crystals. Part I.-Theoretical[END_REF], the flow stress in the gradient microstructure materials can be described as:

0 m p e flow                (6)
0 HP flow k M b d         (7)
where 0

 is the lattice friction stress; HP k is the Hall-Petch slope; d is the grain size; M is the Taylor factor;  is a material constant ;  is the shear modulus; b is the magnitude of Burger's vector of the material ;  is the dislocation density. A modified KME model is employed to represent the evolution of the dislocation density. Moreover, an additional dynamic recovery term is added to decrease the work hardening due to grain refinement:

0 1 2 0 1 20 0 p n e p k d k M k bd b d                                       (

Grain size distribution

As indicated in Section 2.1, the grain size varies from nanometric at the top treated surface to micrometric in the non-affected bulk region. This grain size distribution should be included in the finite element model. Based on experimental results reported in the literature [START_REF] Fang | Revealing Extraordinary Intrinsic Tensile Plasticity in Gradient Nano-Grained Copper[END_REF][START_REF] Wu | Characterization of gradient properties generated by SMAT for a biomedical grade 316L stainless steel[END_REF], an exponential relationship is adopted to describe the variation of grain size in the gradient microstructure:

0 10 d k z d d  (9)
where z is the distance from the treated surface;

  0 lg / / d c g k d d h 
, with g h being the thickness of the gradient grain size layer; d0 and dc are the grain size of the topmost surface and the one of the non-affected bulk region, respectively. The material parameters which describe the gradient grain size are provided in Table 1.

Table 1. Material parameters for gradient grain size [START_REF] Wu | Characterization of gradient properties generated by SMAT for a biomedical grade 316L stainless steel[END_REF].

Symbol d0 (nm) dc (µm) hg (µm) Value 50 20 550

Dislocation density distribution

As presented in Section 2.2, the work hardening induced by SMAT was assessed through measuring FWHM by XRD. As a consequence, the residual accumulated plastic strain p rs  was calculated based on ΔFWHM (Eq. ( 1) and ( 2)). It is well documented that there is a link, to some extent, between accumulated plastic strain and dislocation density, and models were proposed previously in the literature [START_REF] Williamson | Dislocation densities in some annealed and cold-worked metals from measurements on the X-ray debye-scherrer spectrum[END_REF][START_REF] Ghosh | Surface mechanical attrition treatment of additively manufactured 316L stainless steel yields gradient nanostructure with superior strength and ductility[END_REF] to calculate dislocation density based on measured accumulated plastic strain. In this work, the initial dislocation density is related to the residual accumulated plastic strain through the following equation: As shown in Fig. 3, the initial dislocation density (based on Eq. ( 10)) and the accumulated plastic strain decrease gradually from the treated surface to the bulk region. The decrease rate is lower as the depth increases. It can also be seen that this distribution is consistent with the data points presented by Ghosh et al. [START_REF] Ghosh | Surface mechanical attrition treatment of additively manufactured 316L stainless steel yields gradient nanostructure with superior strength and ductility[END_REF]. Fig. 3. In-depth distribution profile of initial dislocation density and accumulated plastic strain. The data points obtained from the literature are also given.

Reconstruction of residual stress field

The residual stress profiles are measured using XRD, as shown in Fig. 2a. These profiles should be implemented in the finite element structure. As presented in previous work [START_REF] Zhou | Reconstruction of residual stress and work hardening and their effects on the mechanical behaviour of a shot peened structure[END_REF], exponential functions based on the stress equilibrium conditions are built to describe the indepth distribution of residual stress:

              max max max 2 - 0 1 2 - 0 1 2 - 0 max 1 e e 1 e 1 i i i l r r rs zz i i t r r rs i i t r r rs i rr i i L L T T T T r r r t                                       (11)
where L0, Li, li, T0, Ti and ti are constant and fitted from the residual stress profiles presented in Fig. 2b. Table 2 lists the values of the parameters involved in Eq. [START_REF] Zhu | Microstructures-based constitutive analysis for mechanical properties of gradient-nanostructured 304 stainless steels[END_REF].

We can obtain the residual stress components distribution, as shown in Fig. 2a. It can be seen that the maximum compressive stress values are at the near-surface region for both axial and circumferential components. It should be noted that tensile residual stress, which occurs in the bulk region, is present to insure the equilibrium of the whole structure.

Finite element simulation

The model with initial variables and fields

A 2D-axisymmetric model is built to analyze the tensile properties of gradient microstructure materials. In order to build a complete model of gradient microstructure materials, the residual stresses are introduced as initial state at the beginning of simulation through the user-defined subroutine SIGINI. The grain size and initial dislocation density are compiled as functions ) implemented into the subroutine UMAT with an elastic-viscoplastic behavior. Material parameters for the model are defined in Table 3. Two analysis steps are performed for the procedure, including a self-equilibrium analysis and followed by a tensile simulation. The residual stress and work hardening distributions are introduced and balanced during the first step. After implementing the finite element model, the distribution of each variable can be verified. It can be observed in Fig. 5a that the yield stress significantly increases with grain size. In the nanostructured layer, the stress level can reach about 1560 MPa, whereas the value in the bulk region is only about 190 MPa. This is due to the fact that the grain boundary strengthening effect can significantly improve the yield strength of the gradient microstructure materials (see Eq. ( 7)). It shows that there is a good agreement between the calculated curve and these experimental points. The implemented grain size distribution can be found in Fig. 5b. The grain size varies from 50 nm at the top surface to 20 µm in the bulk region based on Eq. ( 9). a b Fig. 5. In-depth distribution profiles of (a) yield stress; (b) grain size.

Results and discussion

Mechanical behavior of gradient microstructure materials

In order to compare the local mechanical behavior of the gradient microstructure, four representative depths associated with grain sizes corresponding to 50 nm, 200 nm, 1 µm and 20 µm are chosen to illustrate the stress-strain curves under uniaxial tension loading. For comparison, the simulated stress-strain curve of gradient microstructure with residual stress is also given by dash curves in Fig. 6. It can be observed that the residual stress mainly influences the mechanical behavior in the initial stage of deformation. Moreover, the presence of compressive residual stresses delays the occurrence of plastic deformation, which is more significant in the nanocrystallized layer.

Evolution of dislocation density with strain

Fig. 7 shows the evolution of equivalent plastic strain and dislocation density during tensile loading for the materials with different grain sizes, i.e. at different depths. It can be found that the equivalent plastic strain increases with the imposed strain for all the grain sizes, i.e. in the whole structure (both SMAT affected and bulk regions). Note that the equivalent plastic strain remains constant during the elastic deformation stage. Except for the nanocrystallized layer (Fig. 7d), the dislocation density increases and its increase can be associated with the equivalent plastic strain (see Fig. 7a-c). The phenomenon of dislocation density reduction shown in Fig. 7d can be attributed to the dynamic recovery of dislocations due to work hardening effect in the nanocrystallized layer, as mention in Eq. [START_REF] Yin | Strain rate sensitivity of the ultrastrong gradient nanocrystalline 316L stainless steel and its rate-dependent modeling at nanoscale[END_REF].

By referring to the residual stress profiles presented in Fig. 2a, it can be seen that the residual stress has a small influence on the evolution of equivalent plastic strain and dislocation density (see dashed curves in Fig. 7).

Conclusions

In this study, a dislocation density-based model is adapted to investigate the tensile behavior of gradient microstructure materials by taking into account the residual stress and the grain size. Based on the obtained results, the following conclusions can be drawn:

 By comparing the tensile stress-strain curves of gradient materials with different grain sizes (i.e. at different depths), it can be found that the residual stress plays a significant role in the initial stage of deformation. Moreover, the compressive residual stress delays the occurrence of plastic deformation especially in the very near surface region (with grain sizes of 50 nm and 200 nm).  In the region far from the surface (with grain sizes of 1 µm and 20 µm), the residual stress has a small influence on the evolution of equivalent plastic strain and dislocation density, whereas in the very near surface region, this influence is more pronounced.

Fig. 1 .

 1 Fig. 1. Microstructure observation on the cross-section of the SMATed specimen.

Fig. 6 .Fig. 7 .

 67 Fig. 6. The simulated stress-strain curves of different grain sizes with and without residual stress (RS). a b

Table 2 .

 2 The parameters of residual stress reconstruction model.

	Axial	Circumferential
	L1 (MPa) l1 L2 (MPa) l2 L0 (MPa) T1 (MPa) t1 T2 (MPa) t2 T0 (MPa)
	10547.64 6.95 -11303.32 6.19 188.58 315.23 17.36 -832.24 2.84	91.60

Table 3 .

 3 Parameters for the model of gradient microstructure materials.

	Symbol	σ0 (MPa) KHP (MPa⋅m 1/2 )	M	Α	μ (GPa)	b (nm)
	Value	70 [1]	0.25 [1]	3.06 [9]	0.3 [9]	82	0.256 [9]
	Symbol	k0	k1	k20	n0	ϵ0 (s -1 )	de (μm)
	Value	0.1 [9]	0.026	2.5	21.25 [9] 1 [9]	20 [7]

Acknowledgements

Z. Guo greatly appreciates the financial support from the China Scholarship Council (CSC).


Alessandra Mathias

Nelson Maculan

Elias Adilson

Xavier

Ao Prof

Pereira Abilio

Lucena De

Filho

Ao Prof Juan

Pablo Luna

Aos Professores

Laura Bahiense

Franklin De Lima Marquezino

Luis Felipe

Mag- Alhães De Moraes

Gelabert Luidi

Simonetti

Glaydston Mattos

Laura Silva

Bahiense Da

Silva Leite

José Mario Martínez Pérez

Juan Pablo

Cajahuanca Luna

Sérgio Granville

O Algoritmo

Lagrangiano Hiperbólico

Aumentado Lennin

Mallma Ramirez

Filho Adilson

Elias Xavier

Mallma Lennin

Ramirez

THE HYPERBOLIC AUGMENTED LAGRANGIAN ALGORITHM

The hyperbolic augmented Lagrangian algorithm (HALA) is introduced in the area of continuous optimization for solving nonlinear programming problems. Under mild assumptions, such as: convexity, Slater's qualication and dierentiability, the convergence of the proposed algorithm is proved. We also study the duality theory for the case of the hyperbolic augmented Lagrangian function. Finally, in order to illustrate the algorithm, we present some computational experiments.

Chapter 1 Introduction

We are interested in the nonlinear programming problem subject to inequality constraints, as follows:

min {f (x) | x ∈ S} , (1.1) where S = {x ∈ IR n | g i (x) ≥ 0, i = 1, ..., m} , f and g i , i = 1, ..., m are real-valued functions dened on IR n .

There are a wide variety of methods that solve the problem (1.1), some of them are: the gradient projection method (see [START_REF] Rosen | The Gradient Projection Method for Nonlinear Programming. Part I. Linear Constraints[END_REF]) and the feasible direction methods (see [START_REF] Zoutendijk | Some algorithms based on the principle of feasible directions[END_REF]).

For a better idea of these methods, see [START_REF] Minoux | Mathematical Programming: Theory and Algorithms[END_REF].

On the other hand, methods that also solve the problem (1.1) are for example: the barrier methods, where the logarithmic barrier function (LBF), l(x, r k ) = f (x) -r k m i=1 ln(g i (x)), r k > 0, (1.2) or also the inverse barrier function (IBF),

I(x, r k ) = f (x) + r k m i=1 1 g i (x)
, r k > 0, (1.3) is used (see [START_REF] Fiacco | Nonlinear Programming: Sequential Unconstrained Minimization Techniques[END_REF]). The penalty methods (see [START_REF] Boukari | Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993[END_REF]) and mixed interior-exterior penalty method (see [START_REF] Fiacco | Nonlinear Programming: Sequential Unconstrained Minimization Techniques[END_REF]) also have an important role to solve the problem (1.1). An interesting work where dierent penalty functions are studied computationally and the work of Birgin et al [START_REF] Birgin | Numerical comparison of Augmented Lagrangian algorithms for nonconvex problems[END_REF].

The methodology of the augmented Lagrangians or also called the Lagrange multiplier methods also solve the problem (1.1). The idea of these methods is to convert the constrained problem into a sequence of unconstrained problems. In [START_REF] Bertsekas | Multiplier methods: a survey[END_REF], the advantages of using the multiplier methods over the penalty methods are shown.

When the problem (1.1) has convexity hypothesis, that is, the functions f and -g i , i = 1, ..., m, are convex, there are a variety of augmented Lagrangian methods that solve this problem, some of them are:

(1) The augmented Lagrangian corresponding to the class of function ϕ : [START_REF] Kort | A new penalty function method for constrained minimization[END_REF], see [START_REF] Kort | A new penalty function method for constrained minimization[END_REF]. The subproblem of this augmented Lagrangian algorithm is as follows: Let ϕ : IR → IR,

min x∈I R n f (x) + m i=1 λ k i ϕ ϵ (p i (x)) , (1.4)
the multipliers are updated as follows

λ k+1 i = λ k i ϕ ′ ϵ p i (x k ) , i = 1, ..., m, (1.5)
where p i (x) = -g i (x), i = 1, ..., m, ϵ > 0, ϕ ϵ (t) = ϵϕ t ϵ and ϕ satises 5 properties.

For more details see [START_REF] Kort | A new penalty function method for constrained minimization[END_REF].

(2) Proximal Point and Augmented Lagrangian Methods: [START_REF] Rockafellar | The multiplier method of Hestenes and Powell applied to convex programming[END_REF], see [START_REF] Rockafellar | The multiplier method of Hestenes and Powell applied to convex programming[END_REF] and also see Iusem [START_REF] Iusem | Entropy-like proximal methods in convex programming[END_REF]. The subproblem of this augmented Lagrangian algorithm is as follows:

min x∈I R n f (x) + r k m i=1 max 0, λ k i + h i (x) 2r k 2 -λ k i 2 + r k x -x k 2 , (1.6)
the multipliers are updated as follows

λ k+1 i = max 0, λ k i + h i (x) 2r k , i = 1, ..., m, (1.7)
where h i (x) = -g i (x), i = 1, ..., m, r k ⊂ [r, r] for r ≥ r > 0.

(3) The augmented Lagrangian corresponding to the class of function p ∈ P I : [START_REF] Kort | Combined primal-dual and penalty methods for convex programming[END_REF], see [START_REF] Kort | Combined primal-dual and penalty methods for convex programming[END_REF]. The subproblem of this augmented Lagrangian algorithm is as follows: Let p : IR 2 → IR, this function is continuous dierentiable on IR × (0, +∞),

min x∈I R n f (x) + r k m i=1 p h i (x) r k , λ i , (1.8)
the multipliers are updated as follows

λ k+1 i = ∇ 1 p h i (x k ) r k , λ k i , i = 1, ..., m, (1.9)
where h i (x) = -g i (x), i = 1, ..., m, r k > 0. The function p satises 8 properties.

For more details see [START_REF] Kort | Combined primal-dual and penalty methods for convex programming[END_REF].

(4) The Quadratic Augmented Lagrangian Method: [START_REF] Rockafellar | The multiplier method of Hestenes and Powell applied to convex programming[END_REF], see [START_REF] Rockafellar | The multiplier method of Hestenes and Powell applied to convex programming[END_REF],

Hestenes [START_REF] Hestenes | Multiplier and gradient methods[END_REF] and Powell [START_REF] Powell | A method for nonlinear constraints in minimization problems. Optimization[END_REF]. The subproblem of this augmented Lagrangian algorithm is as follows:

x k ∈ argmin x∈I R n f (x) + 1 2c k m i=1 max 0, λ k-1 i + c k p i (x) 2 , (1.10)
the multipliers are updated as follows λ k i = max 0, λ k-1 i + c k p i (x k ) , i = 1, ..., m.

( 1.11) where p i (x) = -g i (x) ≤ 0, c k > 0.

A solution of the problem (1.1) subject to equality constraints is proposed in [START_REF] Hestenes | Multiplier and gradient methods[END_REF] and [START_REF] Powell | A method for nonlinear constraints in minimization problems. Optimization[END_REF]. Later, the Hestenes-Powell formulation was adapted for the nonlinear programming problem subject to inequality constraints (see [START_REF] Rockafellar | The multiplier method of Hestenes and Powell applied to convex programming[END_REF]). This adaptation denes an augmented Lagrangian function without continuous second derivatives.

This new formulation is known as Hestenes-Powell-Rockafellar augmented Lagrangian function. This function had a very important role to construct a new augmented Lagrangian function, which is continuously dierentiable (see [START_REF] Di Pillo | An augmented Lagrangian function with improved exactness properties[END_REF]).

On the other hand, in [START_REF] Kort | Combined primal-dual and penalty methods for convex programming[END_REF] and [START_REF] Kiwiel | On the twice dierentiable cubic augmented Lagrangian[END_REF] a twice dierentiable augmented Lagrangian function is proposed. Subsequently, a Lagrangian function of class C ∞ is studied in [START_REF] Auslender | Interior proximal and multiplier methods based on second order homogeneous kernels[END_REF] and [START_REF] Polyak | Log-sigmoid multipliers method in constrained optimization[END_REF].

(5) The Exponential Multiplier Method: Tseng and Bertsekas, 1997, see [START_REF] Tseng | On the convergence of the exponential multiplier method for convex programming[END_REF]. The subproblem of this augmented Lagrangian algorithm is as follows: Let ψ : IR → IR and ψ(t) = e t -1,

x k+1 ∈ argmin x∈I R n f (x) + m i=1 λ k i c k i ψ c k i h i (x) , (1.12)
the multipliers are updated as follows

λ k+1 i = λ k i e c k i h i (x k+1 ) , i = 1, ..., m, (1.13)
where h i (x) = -g i (x) ≤ 0, c k i > 0. Tseng and Bertsekas study the exponential multiplier method proposed by [START_REF] Kort | A new penalty function method for constrained minimization[END_REF].

The authors Tseng and Bertsekas study two rules for choosing the penalty parameters and guarantentee the convergence in the ergodic sense. Other works, where the convergence is studied in an ergodic sense, are [START_REF] Iusem | Entropy-like proximal methods in convex programming[END_REF], [START_REF] Jensen | The convergence of a modied barrier method for convex programming[END_REF], [START_REF] Kiwiel | Proximal minimization methods with generalized bregman function[END_REF] and [START_REF] Polyak | Nonlinear rescaling and proximal-like methods in convex optimization[END_REF].

(6) Log-Sigmoid Multiplier Method: Polyak, Griva and Sobieszczanski-Sobieski, 1998, see [START_REF] Polyak | The Newton log-sigmoid method in constrained optimization[END_REF] and [START_REF] Polyak | Log-sigmoid multipliers method in constrained optimization[END_REF], see [START_REF] Polyak | Log-sigmoid multipliers method in constrained optimization[END_REF]. The subproblem of this augmented Lagrangian algorithm is as follows:

x k+1 = argmin x∈I R n f (x) + 2β -1 m i=1 λ i ln 1 + e -βg i (x) -2β -1 m i=1 λ i ln 2 , (1.14)
the multipliers are updated as follows

λ k+1 i = 2λ k i 1 + e βg i (x k+1 ) , i = 1, ..., m, (1.15)
where β > 0.

(7) Nonlinear Rescaling Algorithm: Polyak and Teboulle, 1997, see [START_REF] Polyak | Nonlinear rescaling and proximal-like methods in convex optimization[END_REF]. The subproblem of this augmented Lagrangian algorithm is as follows: Let ψ be a C 2 on the interval (a, +∞), -∞ ≤ a < 0, where ψ(a) = -∞ and ψ ′ (a) = +∞.

x k+1 ∈ argmin x∈I R n f (x) -u -1 m i=1 λ k i ψ (ug i (x)) , (1.16)
the multipliers are updated as follows

λ k+1 i = λ k i ψ ′ ug i (x k+1 ) , i = 1, ..., m, (1.17)
where u > 0, this function ψ satises 6 properties. For more details, see [START_REF] Polyak | Nonlinear rescaling and proximal-like methods in convex optimization[END_REF].

Roman Polyak ([59]) modies functions (1.2) and (1.3) as follows: the modied Frisch function, F (x, λ, r) :

IR n × IR m + × IR + → IR, F (x, λ, r) = f (x) - 1 r m i=1 λ i ln(r g i (x) + 1), (1.18)
and the modied Carroll function, C(x, λ, r) :

IR n × IR m + × IR + → IR, C(x, λ, r) = f (x) + 1 r m i=1 λ i 1 (r g i (x) + 1) -1 .
(

Thus, with the functions (1.18) and (1.19) augmented Lagrangian algorithms are studied, see [START_REF] Polyak | Modied barrier functions: theory and methods[END_REF] and [START_REF] Jensen | The convergence of a modied barrier method for convex programming[END_REF]. Finally, other algorithms that solve convex problems are: [START_REF] Sagastizábal | An infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a lter[END_REF], [START_REF] Gonzaga | A nonlinear programming algorithm based on noncoercive penalty functions[END_REF], [START_REF] Humes | Some inexact hybrid proximal augmented Lagrangian algorithm[END_REF], [START_REF] Necoara | Linear convergence of rst order methods for non-strongly convex optimization[END_REF] and [START_REF] Pinheiro | Solving large-scale reactive optimal power ow problems by a primal-dual M 2 BF approach[END_REF].

Hyperbolic Methodology

This methodology has been developed since previous decades, as follows:

• Xavier, 1982: Master's dissertation ( [START_REF] Xavier | Penalização Hiperbólica-Um Novo Método para Resolução de Problemas de Otimização (master's thesis[END_REF]), Penalização Hiperbólica: Um Novo Método para Resolução de Problemas de Otimização , advisor: João Lizardo Rodrigues Hermes de Araujo.

In [START_REF] Xavier | Penalização Hiperbólica-Um Novo Método para Resolução de Problemas de Otimização (master's thesis[END_REF] the hyperbolic penalty algorithm (HPA) is proposed, it is also studied in [START_REF] Xavier | Penalização Hiperbólica. Anais do I Congresso Latino Americano de Pesquisa Operacional e Engenharia de Sistemas[END_REF], [START_REF] Xavier | Extrapolação em Penalização Hiperbólica, II Congreso Latino Americano de Investigación Operativa e Ingenieria de Sistemas (trabalhos seleccionados)[END_REF], [START_REF] Xavier | Hyperbolic penalty: a new method for nonlinear programming with inequalities[END_REF] and [START_REF] Pereira | A hyperbolic penalty lter method for semiinnite programming[END_REF].

• Xavier, 1992: Doctoral thesis, Penalização Hiperbólica ( [START_REF] Xavier | Penalização Hiperbólica[END_REF]), advisor: Nelson Maculan Filho.

The HPA induces a new augmented Lagrangian algorithm, called HALA-1992 (see [START_REF] Xavier | Penalização Hiperbólica[END_REF]). The characteristic of HALA-1992 is that it considers the updating of the penalty parameter. With this characteristic of HALA-1992 and under a set of assumptions about the problem (1.1), then in that way the rst ideas are given to guarantee the convergence of HALA-1992. In other words, convergence was not guaranteed in this work.

• Xavier, 2010: paper ( [START_REF] Xavier | The hyperbolic smoothing clustering method[END_REF]), The Hyperbolic Smoothing Clustering Method.

Some applications of this method are the following works: [START_REF] Xavier | Optimal covering of plane domains by circles via hyperbolic smoothing[END_REF], [START_REF] Bagirov | Hyperbolic smoothing function method for minimax problems[END_REF], [START_REF] Xavier | Solving the continuous multiple allocation p-hub median problem by the hyperbolic smoothing approach[END_REF], [START_REF] Bagirov | An incremental clustering algorithm based on hyperbolic smoothing[END_REF], [START_REF] Yilmaz | Generalization of hyperbolic smoothing approach for nonsmooth and non-Lipschitz functions[END_REF] and [START_REF] Xavier | Solving the minimum sum-of-squares clustering problem by hyperbolic smoothing and partition into boundary and gravitational regions[END_REF].

HALA E vai entrando na Otimização Matemática

In this occasion, unlike HALA-1992, we consider the xed penalty parameter and we also consider the convexity assumption. So, we propose in this work a new algorithm, which henceforth we will call HALA.

The main contribution of our work is to have guaranteed a basic existence result and the convergence of HALA. In order for us to guarantee the convergence of the algorithm proposed in this work, we use the following classic assumptions, which are widely used in the literature, such as the Slater constraint qualication and convexity. Some works that consider these assumptions are [START_REF] Auslender | Interior proximal and multiplier methods based on second order homogeneous kernels[END_REF], [START_REF] Ben-Tal | Penalty/Barrier multiplier methods for convex programming problems[END_REF] and [START_REF] Polyak | Nonlinear rescaling and proximal-like methods in convex optimization[END_REF]. On the other hand, currently dierent algorithms consider convexity assumptions, see: [START_REF] Goldfarb | Fast alternating linearization methods for minimizing the sum of two convex functions[END_REF], [START_REF] Goldstein | Fast alternating direction optimization methods[END_REF], [START_REF] She | Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization[END_REF], [START_REF] Lan | Iteration-complexity of rst-order augmented Lagrangian methods for convex programming[END_REF], [START_REF] Deng | On the global and linear convergence of the generalized alternating direction method of multipliers[END_REF] and [START_REF] Hong | On the linear convergence of the alternating direction method of multipliers[END_REF]. Therefore, after about 30 years we guarantee an existence result and a convergence result for HALA.

The thesis is organized as follows: In Chapter 2 we present some basic results, we also present HPA and some of its properties. In Chapter 3 we present the hyperbolic augmented Lagrangian function and the HALA. We also study some characteristics of this algorithm. We guarantee the convergence of the HALA and computational results are illustrated. In Chapter 4 The duality theory is studied for the case of the augmented hyperbolic Lagrangian function. In Chapter 5 we give some conclusions of our work. In Chapter 6 we propose some future work.

Chapter 2

Preliminaries

Throughout this thesis we are interested in studying the following optimization problem

(P ) x * ∈ X * = argmin{f (x) | x ∈ S}, where S = {x ∈ IR n | g i (x) ≥ 0, i = 1, ..., m},
is the convex feasible set of the problem (P) and where the function f : IR n → IR is convex, g i : IR n → IR, i = 1, ..., m, are concave functions, we assume that f, g i are continuously dierentiable. Thus (P) is a convex optimization problem. So (P) will be called as the primal problem. We consider the following assumptions.

C1.

The optimal set X * is nonempty, closed, bounded and, consequently, compact.

C2.

Slater constraint qualication holds, i.e., there exists x ∈ S which satises g i (x) > 0, i = 1, ..., m.

A consequence of C1 (see the Theorem 24 and Corollary 20 of [START_REF] Fiacco | Nonlinear Programming: Sequential Unconstrained Minimization Techniques[END_REF]) is that the level set {x ∈ S | f (x) ≤ β} remains bounded for any value β. The C2 assumption guarantees that the interior of S set is nonempty. The condition C1 also imply the existence of a nite vector x * and a number f * such that f (

x * ) = f * = inf S f (x) = min S f (x).
The Lagrangian function of the problem (P) is L :

IR n × IR m + → IR, dened as L(x, λ) = f (x) - m i=1 λ i g i (x), (2.1)
where, λ i ≥ 0, i = 1, ..., m, are called dual variables or Lagrange multipliers. Since the problem (P) is convex, we know that due to assumption C2, the following results will occur: there exists λ * = (λ * 1 , ..., λ * m ), such that, the Karush-Kuhn-Tucker (KKT) conditions hold true, i.e.,

∇ x L(x * , λ * ) = ∇f (x * ) - m i=1 λ * i ∇g i (x * ) = 0, (2.2)
λ * i g i (x * ) = 0, i = 1, ..., m, (2.3)
g i (x * ) ≥ 0, i = 1, ..., m, (2.4)
λ * i ≥ 0, i = 1, ..., m.

(

Moreover, the set of optimal Lagrange multipliers λ * is denoted by

Λ * = λ ∈ IR m + | ∇f (x * ) - m i=1 λ i ∇g i (x * ) = 0, x * ∈ X * ,
it is known that Λ * is a bounded set (and hence compact set) due to C2. The dual function Φ : IR m + → IR, is dened as follows

Φ(λ) = inf x∈I R n L(x, λ), (2.6)
and the dual problem consists of nding

(D) λ ∈ Λ * = argmax{Φ(λ) | λ ∈ IR m + }.

Hyperbolic Penalty

The hyperbolic penalty is meant to solve the problem (P). The penalty method adopts the hyperbolic penalty function (HPF) -λy -(λy) 2 + τ 2 < 0 < -λy + (λy) 2 + τ 2 .

P (y, λ, τ ) = -λy + (λy) 2 + τ 2 , (2.7)
(2.8)

From (2.8), we get that, P (y, λ, τ ) > 0.

Remark 2.1.2 The HPF is originally proposed in [START_REF] Xavier | Penalização Hiperbólica-Um Novo Método para Resolução de Problemas de Otimização (master's thesis[END_REF] and studied in [START_REF] Xavier | Hyperbolic penalty: a new method for nonlinear programming with inequalities[END_REF]. In these studies, the following properties are important for HPF:

(a) P (y, λ, τ ) is asymptotically tangent to the straight lines r 1 (y) = -2λy and r 2 (y) = 0 for τ > 0. P (y, λ, 0) = 0, for y ≥ 0.

P (y, λ, 0) = -2λy, for y < 0.

Due to the properties (a) and (b) the HPF is equivalent to a smoothing of the penalty studied by Zangwill, in [START_REF] Zangwill | Non-linear programming via penalty function[END_REF].

In particular we only use the following properties of the HPF (which are also studied in [START_REF] Xavier | Penalização Hiperbólica-Um Novo Método para Resolução de Problemas de Otimização (master's thesis[END_REF]): P0) P (y, λ, τ ) is k-times continuously dierentiable for any positive integer k for τ > 0.

P1) P (y, λ, τ ) is convex function of y, i.e., ∇ 2 yy P (y, λ, τ ) =

λ 2 τ 2 ((λy) 2 + τ 2 ) 3 2 > 0,
for τ > 0 and λ > 0.

P2) P (y, λ, τ ) is strictly decreasing function of y, i.e.,

∇ y P (y, λ, τ ) = -λ 1 - λy (λy) 2 + τ 2 < 0,
for τ > 0 and λ > 0.

P3) P (0, λ, τ ) = τ, for τ > 0 and λ ≥ 0.

Chapter 3

Hyperbolic Augmented Lagrangian

We dene the Hyperbolic Augmented Lagrangian Function (HALF) of problem (P) by

L H : IR n × IR m ++ × IR ++ → IR, L H (x, λ, τ ) = f (x) + m i=1 P (g i (x), λ i , τ ) = f (x) + m i=1 -λ i g i (x) + (λ i g i (x)) 2 + τ 2 , (3.1)
where τ > 0 is the penalty parameter. Note that this function belongs to class C ∞ if the involved functions f (x) and g i (x), i = 1, ..., m, are too. On the other hand, a variation of (3.1) is proposed and studied in the work of [START_REF] Costa | An articial sh swarm algorithm based hyperbolic augmented Lagrangian method[END_REF] and [START_REF] Rocha | A shifted hyperbolic augmented Lagrangian-based articial sh two-swarm algorithm with guaranteed convergence for constrained global optimization[END_REF].

By comparing (2.1) and (3.1), we see that the function L H may be put in the form

L H (x, λ, τ ) = L(x, λ) + m i=1 (λ i g i (x)) 2 + τ 2 . (3.2)
Analysis of expression (3.2) allows us to see that the modied objective function associated with the hyperbolic penalty may be decomposed as the sum of the Lagrangian function along with a summation of terms containing squares of the products between the values of the constraints and their corresponding multipliers (complementary slacks). We are aware that at any optimal point (x * , λ * ) we must have λ * i g i (x * ) = 0, i = 1, ..., m, and therefore at this point the summation takes on a minimum value equal to m i=1 τ = mτ.

From this point of view the summation in expression (3.2) may be interpreted as a penalty for the noncompliance with the condition of complementarity of the slacks which is added to the Lagrangian function. In the composition of the modied objective function, when we attempt to minimize this portion, we will automatically be seeking the optimal solution where equalities λ * i g i (x * ) = 0, i = 1, ..., m prevail. Now, let us consider the following assumption:

C3. For every τ > 0 and λ > 0. Also for every l < ∞, the level set

M = {x ∈ IR n | L H (x, λ, τ ) ≤ l} , is bounded.
Remark 3.0.1 We know that the function P is convex by P1). Now the assumption C3 is veried if in particular the function f is strongly convex in x. That way L H will also be strongly convex in x. Other works that also use the strong convexity assumptions are:

[2], [START_REF] Kiwiel | Proximal minimization methods with generalized bregman function[END_REF], [START_REF] Bertsekas | On the Method of Multipliers for Convex Programming[END_REF], [START_REF] Silva | Rescaling and stepsize selection in proximal methods using separable generalized distances[END_REF], [START_REF] Rockafellar | Augmented Lagrangian and applications of the proximal point algorithm in convex programming[END_REF], [START_REF] Kang | Inexact accelerated augmented Lagrangian methods[END_REF], [START_REF] Sabach | Faster Lagrangian-based methods in convex optimization[END_REF], [START_REF] Silva | Rescaling and stepsize selection in proximal methods using separable generalized distances[END_REF] and [START_REF] Xu | Iteration complexity of inexact augmented lagrangian methods for constrained convex programming[END_REF]. See also [START_REF] Goldfarb | A modied barrieraugmented Lagrangian method for constrained minimization[END_REF].

We present the HALA to solve the nonlinear problem (P).

Algorithm HALA

Step 1. Let k := 0 (initialization). Take initial values λ 0 = (λ 0 1 , ..., λ 0 m ) ∈ IR m ++ , τ ∈ IR ++ .

Step 2. Solve the unconstrained minimization problem (primal update):

x k+1 ∈ argmin x∈I R n L H (x, λ k , τ ) = argmin x∈I R n f (x) + m i=1 -λ k i g i (x) + λ k i g i (x) 2 + τ 2 .
Step 3. Updating of Lagrange multipliers (dual update):

λ k+1 i = λ k i 1 - λ k i g i (x k+1 ) (λ k i g i (x k+1 )) 2 + τ 2 , i = 1, ..., m. (3.3)
Step 4. If the pair (x k+1 , λ k+1 ) satises the stopping criteria: Then Stop.

Step 5. k := k + 1. Go to Step 2.

HALA considers an initial vector λ 0 > 0 and τ > 0. Note that HALA assumes a xed value for the penalty parameter τ. Considering a xed penalty parameter can also be observed in the following studies [START_REF] Jensen | The convergence of a modied barrier method for convex programming[END_REF], [START_REF] Kort | A new penalty function method for constrained minimization[END_REF], [START_REF] Kort | Multiplier method for convex programming[END_REF] and [START_REF] Rockafellar | The multiplier method of Hestenes and Powell applied to convex programming[END_REF]. With this information, the HALA generate the primal sequence in Step 2 and the multiplier estimates in Step 3.

In Step 4, we can consider dierent stopping criteria. For example, we can consider some of the following criteria studied in [START_REF] Breitfeld | Computational experience with penalty-barrier methods for nonlinear programming[END_REF]:

-min i=1,...,m

g i (x k ) < β and f (x k ) -f (x k-1 ) 1 + |f (x k-1 )| < 10 -2 β, or max -min i=1,...,m g i (x k ), m i=1 λ k i g i (x k ) 1 + ∥x k ∥ 2 , ∇f (x k ) -m i=1 λ k i ∇g i (x k ) ∞ 1 + ∥x k ∥ 2 < β,
where β > 0.

Notice that HALA is based in the exact unconstrained minimization of the HALF. In [START_REF] Kort | Combined primal-dual and penalty methods for convex programming[END_REF] an exact unconstrained minimization of the augmented Lagrangian is also discussed, also see [START_REF] Ben-Tal | Penalty/Barrier multiplier methods for convex programming problems[END_REF].

Study of the HALA

By C3, hence there exists x k+1 ∈ IR n such that

L H (x k+1 , λ k , τ ) = min x∈I R n L H (x, λ k , τ ), thus ∇ x L H (x k+1 , λ k , τ ) = 0 holds, i.e., ∇f (x k+1 ) - m i=1 λ k i 1 - λ k i g i (x k+1 ) (λ k i g i (x k+1 )) 2 + τ 2 ∇g i (x k+1 ) = 0, (3.4)
substituting (3.3) in (3.4), we have

∇ x L H (x k+1 , λ k , τ ) = ∇f (x k+1 ) - m i=1 λ k+1 i ∇g i (x k+1 ) = ∇ x L(x k+1 , λ k+1 ) = 0, (3.5)
for any τ > 0.

We observe that x k+1 and λ k+1 satisfy ∇ x L(x k+1 , λ k+1 ) = 0, shows that x k+1 is the minimizer of L(x, λ k+1 ) (i.e., x k+1 attains the minimum in (2.6)), i.e.,

Φ(λ k+1 ) = L(x k+1 , λ k+1 ) = min x∈I R n L(x, λ k+1 ) and λ k+1 ∈ IR m ++ , thus, it follows that Φ(λ k+1 ) = f (x k+1 ) - m i=1 λ k+1 i g i (x k+1 ). (3.6) From (3.6) we obtain -g(x k+1 ) = -g 1 (x k+1 ), • • • , -g m (x k+1 ) T ∈ ∂Φ(λ k+1 ),
where ∂Φ(λ k+1 ) is the subdierential of Φ(λ) at λ = λ k+1 .

In the following remark, we analyze what happens with Lagrange multipliers (iteration

(3.
3)) depending on the type of restriction we have. First, for x ∈ IR n , we dene the following sets of indices

I 0 = {i ∈ {1, ..., m} | g i (x) = 0} , I -= {i ∈ {1, ..., m} | g i (x) < 0}
and

I + = {i ∈ {1, ..., m} | g i (x) > 0} , such that I 0 ∩ I + = ∅, I 0 ∩ I -= ∅, I + ∩ I -= ∅ and I 0 ∪ I + ∪ I -= {1, ..., m} .
Remark 3.2.1 Let {λ k } be a sequence generated by HALA such that λ k i > 0, i = 1, ..., m and let τ > 0 xed. Let us consider the following cases: (c1) If i ∈ I 0 , then we have at the k-th iteration g i (x k+1 ) = 0, then by (3.3), we get,

λ k+1 i = λ k i .
We also obtain:

λ k i -λ k+1 i g i (x k+1 ) = 0, ∀i ∈ I 0 .
(c2) If i ∈ I + , then we have at the k-th iteration g i (x k+1 ) > 0.

So we can get,

λ k i g i (x k+1 ) > 0, λ k i g i (x k+1 ) λ k i g i (x k+1 ) 2 + τ 2 > 0, 1 > 1 - λ k i g i (x k+1 ) λ k i g i (x k+1 ) 2 + τ 2 , λ k i > λ k i   1 - λ k i g i (x k+1 ) λ k i g i (x k+1 ) 2 + τ 2   , (3.7)
then by (3.3) in (3.7), we get, λ k i > λ k+1 i . We also obtain:

λ k i -λ k+1 i g i (x k+1 ) > 0, ∀i ∈ I + .
(c3) If i ∈ I -, then we have at the k-th iteration g i (x k+1 ) < 0.

So we can get,

λ k i g i (x k+1 ) < 0, λ k i g i (x k+1 ) λ k i g i (x k+1 ) 2 + τ 2 < 0, 1 < 1 - λ k i g i (x k+1 ) λ k i g i (x k+1 ) 2 + τ 2 , λ k i < λ k i   1 - λ k i g i (x k+1 ) λ k i g i (x k+1 ) 2 + τ 2   , (3.8)
then by (3.3) in (3.8), we get, λ k i < λ k+1 i . We also obtain:

λ k i -λ k+1 i g i (x k+1 ) > 0, ∀i ∈ I -.
Of the three previous cases, we can note that we have the following

λ k i -λ k+1 i g i (x k+1 ) ≥ 0, i = 1, ..., m.
In the following result, we will demonstrate the positivity of the updated Lagrange multipliers.

Proposition 3.2.1 Let λ k = (λ k 1 , ..., λ k m ) | k = 1, 2, ... ⊂ IR m . If λ k ∈ IR m ++ then λ k+1 ∈ IR m ++ , i = 1, ..., m.
Proof. Let τ > 0 be xed. Since we have 0 < τ 2 , we can obtain the following

λ k i g i (x k+1 ) 2 < λ k i g i (x k+1 ) 2 + τ 2 , i = 1, ..., m, so, λ k i g i (x k+1 ) < λ k i g i (x k+1 ) 2 + τ 2 , i = 1, ..., m, thus, -λ k i g i (x k+1 ) 2 + τ 2 < λ k i g i (x k+1 ) < λ k i g i (x k+1 ) 2 + τ 2 , i = 1, ..., m,
from this, we can get

-1 < λ k i g i (x k+1 ) λ k i g i (x k+1 ) 2 + τ 2 < 1, i = 1, ..., m,
from the latter it follows that

0 < λ k i   1 - λ k i g i (x k+1 ) λ k i g i (x k+1 ) 2 + τ 2   < 2λ k i , i = 1, ..., m, (3.9)
then from the expression above and by (3.3), we get that, λ k+1 i > 0, i = 1, ..., m. Proof. From the concavity of Φ(•) and since -g(x k+1 ) ∈ ∂Φ(λ k+1 ), we obtain

Φ(λ) -Φ(λ k+1 ) ≤ -g(x k+1 ) λ -λ k+1 , (3.11)
now by rewriting (3.11) and considering λ = λ k , we obtain

Φ(λ k+1 ) -Φ(λ k ) ≥ g(x k+1 ) λ k -λ k+1 , (3.12)
writing again (3.12), so we have the expression

Φ(λ k+1 ) -Φ(λ k ) ≥ m i=1 g i (x k+1 ) λ k i -λ k+1 i . (3.13)
On the other hand, we can rewrite (3.3), as follows,

λ k i -λ k+1 i = λ k i 2 g i (x k+1 ) λ k i g i (x k+1 ) 2 + τ 2 , i = 1, ..., m, (3.14)
this expression (3.14) is replaced on the right side of inequality (3.13), we get

Φ(λ k+1 ) -Φ(λ k ) ≥ m i=1   λ k i g i (x k+1 ) 2 λ k i g i (x k+1 ) 2 + τ 2   ≥ 0, (3.15)
so, we have,

Φ(λ k+1 ) ≥ Φ(λ k ).
λ k ∈ Γ = {λ ∈ IR m + | Φ(λ 0 ) ≤ Φ(λ)} for all k ∈ IN and hence {λ k } is a bounded sequence.
We present a preliminary result which will be used to guarantee the complementarity condition in our algorithm. Proof. By the hypothesis, for ϵ ∈ (0, 1) xed, there exists k 0 ∈ IN , such that

a k √ a k + d < ϵ, ∀k ≥ k 0 , -ϵ < a k √ a k + d < ϵ, ∀k ≥ k 0 , 0 < a k √ a k + d + ϵ < 2ϵ, ∀k ≥ k 0 . (3.16)
On the other hand, we know that 0 < ϵ, then

a k √ a k + d < a k √ a k + d + ϵ, (3.17)
now, we replace (3.17) in (3.16), then we get

a k √ a k + d < a k √ a k + d + ϵ < 2ϵ,
so, from the inequality above we have

a k 2 √ a k + d < ϵ, ∀k ≥ k 0 . (3.18)
Also, on the other hand, we know that

√ a k + d -1 2 ≥ 0, then a k + d + 1 ≥ 2 a k + d, ∀k ≥ k 0 , so, 1 a k + d + 1 ≤ 1 2 √ a k + d , ∀k ≥ k 0 , from (3.18
) and from the previous inequality, we obtain the following

a k a k + d + 1 ≤ a k 2 √ a k + d < ϵ, ∀k ≥ k 0 , (3.19)
then of (3.19), we get

a k a k + d + 1 ≤ ϵ, ∀k ≥ k 0 , a k ≤ a k ϵ + ϵ (d + 1) , ∀k ≥ k 0 , a k -a k ϵ ≤ ϵ (d + 1) , ∀k ≥ k 0 , a k (1 -ϵ) ≤ ϵ (d + 1) , ∀k ≥ k 0 , thus, a k ≤ ϵ(d+1) 1-ϵ
, ∀k ≥ k 0 which implies that lim k→∞ a k = 0.

0 ≤ ∞ k=1 m i=1   λ k i g i (x k+1 ) 2 λ k i g i (x k+1 ) 2 + τ 2   ≤ ∞ k=1 Φ(λ k+1 ) -Φ(λ k ) , we notice that ∞ k=1 Φ(λ k+1 ) -Φ(λ k
) is a convergent series (i.e., the partial sum is bounded above), it follows

0 ≤ ∞ k=1 m i=1   λ k i g i (x k+1 ) 2 λ k i g i (x k+1 ) 2 + τ 2   ≤ lim k→∞ Φ(λ k ) -Φ(λ 1 ) ≤ f * -Φ(λ 1 ) < ∞,
therefore, for the test of comparison, we obtain

lim k→∞ m i=1   λ k i g i (x k+1 ) 2 λ k i g i (x k+1 ) 2 + τ 2   = 0. (3.21)
We note each term in the summation of (3.21) is nonnegative, thus

lim k→∞   λ k i g i (x k+1 ) 2 λ k i g i (x k+1 ) 2 + τ 2   = 0, i = 1, ..., m, (3.22
Φ(λ k+1 ) -Φ(λ k ) ≥ m i=1 g i (x k+1 ) λ k i -λ k+1 i ≥ 0, (3.24)
and by Proposition 3.2.2 we know that {Φ(λ k )} is convergent, so, it follows lim k→∞ Φ(λ k+1 ) -Φ(λ k ) = 0, and so from (3.24) we obtain

lim k→∞ m i=1 g i (x k+1 ) λ k i -λ k+1 i = 0, (3.25) now since g i (x k+1 ) λ k i -λ k+1 i ≥ 0 (by Remark 3.2.

Convergence Result

In this section, we are going to consider the following assumption.

C4. The whole sequence {x k } is convergent to x, where x is assumed a feasible point, i.e., g i (x) ≥ 0, i = 1, ..., m.

Similar to assumption C4 can also be seen in Hartman [START_REF] Hartman | Iterative determination of parameters for an exact penalty function[END_REF], Nguyen and Strodiot [START_REF] Nguyen | On the convergence rate for a penalty function method of exponential type[END_REF], [START_REF] Birgin | The boundedness of penalty parameters in an augmented Lagrangian method with constrained subproblems[END_REF], [START_REF] Fernández | An inexact restoration strategy for the globalization of the sSQP method[END_REF] and [START_REF] Echebest | Convergence results of an augmented Lagrangian method using the exponential penalty function[END_REF]. Finally, we ensure that the subsequence generated by the algorithm HALA converges to a KKT point, but not necessarily the entire sequence. (3.28)

Moreover, passing the limit in (3.5), we obtain

∇ x L(x, λ) = ∇f (x) - m i=1 λi ∇g i (x) = 0.
Thus (x, λ) satises (2.2) -(2.5) for all i = 1, ..., m, hence (x, λ) is a KKT point. Thus x is optimal for the problem (P) and λ is a Lagrange multiplier.

Computational Illustration

The computationally illustrate presented below were obtained with a preliminary Fortran implementation for the HALA. The program were compiled by the GNU Fortran compiler version 4:7.4.0-1ubuntu2.3. The numerical Experiments are conducted on a Notebook with operating system Ubuntu 18.04.5, CPU i7-3632QM and 8GB RAM. The unconstrained minimization tasks were carried out by means of a Quasi-Newton algorithm employing the BFGS updating formula, with the function VA13 from HSL library [START_REF]Harwell Subroutine Library: A collection of Fortran codes for large scale scientic computation[END_REF]. The algorithm stop when the solution is viable (feasible) an the absolute value of the dierence of the two consecutives solutions x k -x k-1 is less than 10 -7 .

Test Problems

In this section we show ve test problems.

Example 3.4.1

min x∈I R 3 f (x) = 2x 2 1 + 3x 2 2 + x 2 3 s.t. g 1 (x) = x 2 1 + x 2 2 + x 2 3 -8 ≤ 0, g 2 (x) = 4x 2 1 -32x 1 + 36x 2 2 -144x 2 + 9x 2 3 -18x 3 + 181 ≤ 0.
Starting with x 0 = (1.5, 1.5, 1) and f (x 0 ) = 12.2500000000. The minimum value is f (x * ) = 11.3792836271 at the optimal solution x * = (1.3958680939, 1.5256305407, 0.7069246891).

Example 3.4.2 Problem 43 (Rosen-Suzuki) of [START_REF] Hock | Test Examples for Nonlinear Programming Codes[END_REF].

min x∈I R 4 f (x) = x 2 1 + x 2 2 + 2x 2 3 + x 2 4 -5x 1 -5x 2 -21x 3 + 7x 4 s.t. g 1 (x) = 8 -x 2 1 -x 2 2 -x 2 3 -x 2 4 -x 1 + x 2 -x 3 + x 4 ≥ 0, g 2 (x) = 10 -x 2 1 -2x 2 2 -x 2 3 -2x 2 4 + x 1 + x 4 ≥ 0, g 3 (x) = 5 -2x 2 1 -x 2 2 -x 2 3 -2x 1 + x 2 + x 4 ≥ 0.
Starting with x 0 = (0, 0, 0, 0) and f (x 0 ) = 0. The minimum value is f (x * ) = -44 at the optimal solution x * = (0, 1, 2, -1).

Example 3.4.3 Problem 11 of [START_REF] Hock | Test Examples for Nonlinear Programming Codes[END_REF].

min x∈I R 2 f (x) = (x 1 -5) 2 + x 2 2 -25 s.t. g 1 (x) = -x 2 1 + x 2 ≥ 0.
Starting with x 0 = (4.9, 0.1) (not feasible) and f (x 0 ) = -24.98. The minimum value is f (x * ) = -8.498464223.

Example 3.4.4 See, Remark 4.2, of [START_REF] Polyak | Log-sigmoid multipliers method in constrained optimization[END_REF].

min x∈I R f (x) = 3x s.t. g 1 (x) = x ≥ 0.
Starting with λ 0 = 1. The optimal solution is X * = {0} and Λ * = {3} .

Example 3.4.5

min x∈I R 4 f (x) = -6x 1 -3x 2 -2x 3 -x 4 s.t. g 1 (x) = x 1 ≥ 0, g 2 (x) = x 2 ≥ 0, g 3 (x) = x 3 ≥ 0, g 4 (x) = x 4 ≥ 0, g 5 (x) = -x 1 + 1 ≥ 0, g 6 (x) = -x 2 + 1 ≥ 0, g 7 (x) = -x 3 + 1 ≥ 0, g 8 (x) = -x 4 + 1 ≥ 0, g 9 (x) = -3x 1 -2x 2 -3x 3 -3x 4 + 6 ≥ 0.
Starting with x 0 = (0.1, 0.1, 0.5, 0.5) and f (x 0 ) = -0.240000000E + 01. The minimum value is f (x * ) = -0.966666667E + 01 at the optimal solution x * = (0.100000000E + 01, 0.100000000E + 01, 0.333333332E + 00, 0.621067930E -09).

Example 3.4.6 See, pag. 84 of [START_REF] Eckstein | A practical general approximation criterion for methods of multipliers based on Bregman distances[END_REF].

min x∈I R f (x) = 0x s.t. g 1 (x) = x ≥ 0.
Example 3.4.7 See, pag. 30 of [START_REF] Iusem | Augmented Lagrangian methods and proximal point methods for convex optimization[END_REF].

min x∈I R f (x) = 1 s.t. g 1 (x) = 1 -e x ≥ 0.

Results

Tables 3.1-3.11 summarize the computational results for these ve problems. For each test problem we present two tables. The rst table contains information about the primal sequence and the second table contains information about the dual sequence. For each table, k is the number of iterations, τ is the penalty parameter, λ is the Lagrange multiplier, x is the primal variable, f (x) is the objective value, g i (x) are the constraints of each example, L H (x, λ, τ ) is the value of the HALF and via = viable = f easible where, in each iteration, the obtained point can be viable, then its value is "0 = yes" or the point can be inviable, then the value is "1 = not".

In Tables 3.1, 3.3, 3.5, 3.7 and 3.9, we reports the optimal solutions, the value of the objective function and the value of the HALF found by our proposed algorithm.

For Problems 3.4.1, 3.4.2, 3.4.3, 3.4.4 and 3.4.5 our algorithm converges to the exact solution within the precision of the computer.

In Tables 3.2, 3.4, 3.6, 3.8, 3.10 and 3.11 we reports the behavior of the multipliers, this issue is studied in Subsection 3.2 of this work. In particular, in Table 3.2 and Table 3.4 we clearly observe that λ k+1 i = 2λ k i , this happens when the solution is not viable and the value of τ is small. In these tables, we can also see the active constraints, for each proposed example. In Table 3.4 our active constraints are g 1 (x) and g 3 (x) as suggested in dierent papers, in particular see [START_REF] Kort | Combined primal-dual and penalty methods for convex programming[END_REF].

In the Example 3.4.3 when we consider the infeasible starting point (4.9, 0.1)

suggested by [START_REF] Hock | Test Examples for Nonlinear Programming Codes[END_REF] our algorithm converges to the optimal solution in 7 iterations. When we consider the initial viable point [START_REF] Akrotirianakis | Globally convergent interior-point algorithm for nonlinear programming[END_REF][START_REF] Auslender | Interior proximal and multiplier methods based on second order homogeneous kernels[END_REF] suggested by us, our algorithm converges in 5 iterations, see Table 3.5. In the Example 3.4.4 when we used the initial value λ 0 = 1 used by [START_REF] Polyak | Log-sigmoid multipliers method in constrained optimization[END_REF] with τ = 1 we observe that our algorithm does not converge. But our algorithm converge when λ 0 = 10 and τ = 1, then the results obtained by HALA are presented in Tables 3.7-3.8. In Problem 3.4.5, we can notice that the primal and dual sequence always remains in the viable region, see Tables 3.9-3.11.

In the Example 3.4.6, we consider the following initial parameters: x 0 = 1, λ 0 = 1 and τ = 1. In this way our algorithm HALA converges to the primal solution and a Lagrange multiplier, see Tables 3.12 -3.13. On the other hand, the augmented Lagrangian algorithm studied by Eckstein also studies the same problem, but to guarantee the convergence of the Lagrange multipliers, he needs a condition on the gradient of the augmented Lagrangian function, for more details of this example see [START_REF] Eckstein | A practical general approximation criterion for methods of multipliers based on Bregman distances[END_REF]. In the Example 3.4.7, we consider the following initial parameters: x 0 = -2, λ 0 = 1 and τ = 1. In this way our algorithm HALA converges to the primal solution and a Lagrange multiplier, see Tables 3.14-3.15. On the other hand, using the generalized augmented Lagrangian method (GALB), convergence is not guaranteed for this example. Now, considering the generalized doubly augmented Lagrangian method (is based on proximal point methods), convergence is guaranteed for this example. For more details of this example see [START_REF] Iusem | Augmented Lagrangian methods and proximal point methods for convex optimization[END_REF].

k x 1 x 2 x 3 x 4 f (x) L H (x, λ, τ )
via 0 0.100000000E+00 0.100000000E+00 0.500000000E+00 0.500000000E+00 -0.240000000E+01 -0.239984689E+01 0 1 0.998900380E+00 0.998287244E+00 0.332446822E+00 0.221619670E-02 -0.965537385E+01 -0.964379954E+01 0 2 0.999350555E+00 0.998842976E+00 0.333095079E+00 0.158967189E-02 -0.966041209E+01 -0.876677236E+01 0 3 0.100000000E+01 0.100000000E+01 0.333333332E+00 0.621067930E-09 -0.966666667E+01 -0.876670427E+01 0 4 0.100000000E+01 0.100000000E+01 0.333333332E+00 0.621067930E-09 -0.966666667E+01 -0.876670427E+01 0

g 1 (x) g 2 (x) g 3 (x) g 4 (x) g 5 (x)
k via Duality Theory for the Hyperbolic Augmented Lagrangian

In this section we are interested in developing the duality theory for HALF in the Euclidean space.

The main result of this section is guarantee the strong duality for HALF for the convex case. In this way we assure a solution to the primal and dual problems. With these results, we can also note that HALF has properties similar to Log-sigmoid Lagrangian function (LSLF), see [START_REF] Polyak | Log-sigmoid multipliers method in constrained optimization[END_REF]; modied Frisch function (MFF) and Modied Carroll function (MCF), these last two functions are studied in [START_REF] Polyak | Modied barrier functions: theory and methods[END_REF].

Proposition 4.0.1 Let us assume that if f (x) and all g i (x) ∈ C 2 and that f (x) is strictly convex, then L H (x, λ, τ ) is strictly convex in IR n for any xed λ > 0 and τ > 0.

Proof. We only need to prove that the Hessiana of L H is dened positive. Let are λ = (λ 1 , ..., λ m ) > 0 and τ > 0 xed. The Hessian of L H (x, λ, τ ) is In (4.1), the ∇ 2 xx g i (x) is factored, so, we can rewrite (4.1), as follows

∇ 2 xx L H (x, λ, τ ) = ∇ 2 xx f (x) - m i=1 λ i ∇ 2 xx g i (x) + m i=1 (λ i ) 2 (λ i g i (x)) 2 + τ 2 - (λ i ) 4 g 2 i (x) ((λ i g i (x)) 2 + τ 2 ) 3 2 ∇ x g i (x)∇ x g T i (x) + m i=1 (λ i ) 2 g i (x) (λ i g i (x)) 2 + τ 2 ∇ 2 xx g i (x).
∇ 2 xx L H (x, λ, τ ) = ∇ 2 xx f (x) - m i=1 λ i 1 - λ i g i (x) (λ i g i (x)) 2 + τ 2 ∇ 2 xx g i (x) + m i=1 (λ i ) 2 (λ i g i (x)) 2 + τ 2 - (λ i ) 4 g 2 i (x) ((λ i g i (x)) 2 + τ 2 ) 3 2 ∇ x g i (x)∇ x g T i (x). (4.2)
On the other hand, since we have τ 2 > 0, we can get

(λ i g i (x)) 2 + τ 2 > (λ i g i (x)) 2 , (4.3)
now we multiply by λ 2 i in (4.3), so it follows that

(λ i g i (x)) 2 + τ 2 λ 2 i > λ 4 i g 2 i (x),
the above inequality, we can rewrite it as

(λ i g i (x)) 2 + τ 2 3 2 (λ i g i (x)) 2 + τ 2 1 2 λ 2 i > λ 4 i g 2 i (x), so, λ 2 i (λ i g i (x)) 2 + τ 2 1 2 > λ 4 i g 2 i (x) (λ i g i (x)) 2 + τ 2 3 2 , thus, it follows λ 2 i (λ i g i (x)) 2 + τ 2 1 2 - λ 4 i g 2 i (x) (λ i g i (x)) 2 + τ 2 3 2 > 0. (4.4)
We replace (4.4) in (4.2) and since -λ i 1 -

λ i g i (x) √ (λ i g i (x)) 2 +τ 2 < 0, (by P2) in (4.2), we get that, ∇ 2 xx L H (x, λ, τ ) > 0, for λ > 0 and τ > 0 xed.
Recall that strict convexity implies convexity.

Remark 4.0.1 From C3 and Proposition 4.0.1 for any λ > 0 and any τ > 0 there exists a unique minimizer

x = x(λ, τ ) = argmin {L H (x, λ, τ ) | x ∈ IR n }
for problem (P) with the assumption C1.

Duality

In this section, we adapt the classic results already existing in the literature: Chapter 9 of [START_REF] Polyak | Introduction to Optimization[END_REF] and Section 7 of [START_REF] Polyak | Modied barrier functions: theory and methods[END_REF] for our HALF. The following result is also veried by MFF and MCF, see [START_REF] Polyak | Modied barrier functions: theory and methods[END_REF]. (i) There exists a vector λ * ≥ 0 such that

λ * i g i (x * ) = 0, i = 1, ..., m and L H (x, λ * , τ ) ≥ L H (x * , λ * , τ ), ∀x ∈ IR n . (4.5) (ii) The pair (x * , λ * ) is a saddle point of L H , that is, L H (x, λ * , τ ) ≥ L H (x * , λ * , τ ) ≥ L H (x * , λ, τ ), ∀x ∈ IR n , ∀λ ∈ IR m + . (4.6)
Proof. (⇒) Let any τ > 0 xed. Assume x * is a solution for convex problem (P)

satisfying the assumption C2. Then system

f (x) -f (x * ) < 0, -g i (x) < 0, i = 1, ..., m,
has no solution in IR n . Hence, by the Proper Separation Theorem (see, Theorem 2.26 (iv) of Dhara and Dutta [START_REF] Dhara | Optimality conditions in Convex Optimization[END_REF]), there exists a vector ( λ, λ)

̸ = (0, 0) ∈ IR × IR m such that λ (f (x) -f (x * )) - m i=1 λi g i (x) ≥ 0,
for all x ∈ IR n . We rewrite the inequality above as

λ (f (x) -f (x * )) ≥ m i=1 λi g i (x), (4.7)
for all x ∈ IR n . Now, we follow an analysis similar to Theorem 4.2 of [START_REF] Dhara | Optimality conditions in Convex Optimization[END_REF], so by C2, we have that there exists λ * i = λi λ , i = 1, ..., m, with λ > 0. Then, by (4.7) we have

f (x) -f (x * ) ≥ m i=1 λ * i g i (x), (4.8)
for all x ∈ IR n . In particular, (4.8) holds for x = x * . So we get

0 ≥ m i=1 λ * i g i (x * ). (4.9)
On the other hand, since, g i (x * ) ≥ 0 and λ * i ≥ 0 for i = 1, ..., m, then by (4.9) we obtain

λ * i g i (x * ) = 0, i = 1, ..., m, (4.10)
so we have the rst part of (4.5). Now, we are interested in proving the second part of (4.5). From (4.10) and (4.8), we have

f (x * ) - m i=1 λ * i g i (x * ) = f (x * ) ≤ f (x) - m i=1 λ * i g i (x), (4.11)
for all x ∈ IR n . Now, since we have (4.10), also, we can obtain

(λ * i g i (x * )) 2 + τ 2 ≤ (λ * i g i (x)) 2 + τ 2 , i = 1, ..., m, so, we have the following m i=1 (λ * i g i (x * )) 2 + τ 2 ≤ m i=1 (λ * i g i (x)) 2 + τ 2 , (4.12)
adding the expressions (4.11) and (4.12), we get

L H (x, λ * , τ ) ≥ L H (x * , λ * , τ ), ∀x ∈ IR n , (4.13)
in this way, we nish the proof of (4.5).

We are interested in verifying item (ii) now. But, rst we will prove that L H (x * , λ * , τ ) = f (x * ) + mτ. Indeed, by denition of L H , we have On the other hand, as x * is feasible, i.e.,

L H (x * , λ * , τ ) = f (x * ) - m i=1 λ * i g i (x * ) + m i=1 (λ * i g i (x * )) 2 + τ 2 ,
g i (x * ) ≥ 0, i = 1, ..., m. (4.16)
By applying the property P2 of HPF in (4.16), we obtain

P (g i (x * ), λ i , τ ) ≤ P (0, λ i , τ ), i = 1, ..., m. (4.17)
By applying property P3, on the right side of expression (4.17), we will obtain

P (g i (x * ), λ i , τ ) ≤ τ, f or λ i ≥ 0, i = 1, ..., m. (4.18)
By performing the sum of 1 to m in (4.18) it follows

m i=1 P (g i (x * ), λ i , τ ) ≤ m i=1 τ = mτ.
Adding f (x * ) to both sides of the expression, we obtain

f (x * ) + m i=1 P (g i (x * ), λ i , τ ) ≤ f (x * ) + mτ. (4.19)
By denition of L H , (4. [START_REF] Cristofari | An Augmented Lagrangian Method Exploiting an Active-Set Strategy and Second-Order Information[END_REF] becomes

L H (x * , λ, τ ) ≤ f (x * ) + mτ. (4.20)
Now, by (4.20) and (4.15) we have

L H (x * , λ, τ ) ≤ f (x * ) + mτ = L H (x * , λ * , τ ). (4.21)
Finally, from (4.13) and (4.21), there is λ * ≥ 0 such that the primal-dual solution (x * , λ * ) is a saddle point of L H , ∀x ∈ IR n and τ > 0.

(⇐) We assume that (x * , λ * ) is a saddle point of L H , so (4.6) is hold. Then, for all x ∈ IR n , λ ∈ IR m + and for any τ > 0 xed, we have

f (x * ) - m i=1 λ i g i (x * ) + m i=1 (λ i g i (x * )) 2 + τ 2 = L H (x * , λ, τ ) ≤ L H (x * , λ * , τ ) = f (x * ) - m i=1 λ * i g i (x * ) + m i=1 (λ * i g i (x * )) 2 + τ 2 .
λ i g i (x * ) + m i=1 (λ i g i (x * )) 2 + τ 2 ≤ - m i=1 λ * i g i (x * ) + m i=1 (λ * i g i (x * )) 2 + τ 2 , (4.23)
for all λ i ≥ 0, i = 1, ..., m.

This relation (4.23) is possible only if g i (x * ) ≥ 0. Since, if this relation is violated (i.e., g i (x * ) < 0) for some index i, we can choose λ i suciently large such that (4.23) becames false. So, x * is feasible for problem (P).

We will prove the complementarity condition of (4.5). So again, by (4.23), and since that λ i ≥ 0, i = 1, ..., m, in particular taking λ i = 0, i = 1, ..., m, in (4.23), we obtain

m i=1 τ ≤ - m i=1 λ * i g i (x * ) + m i=1 (λ * i g i (x * )) 2 + τ 2 , m i=1 τ + m i=1 λ * i g i (x * ) ≤ m i=1 (λ * i g i (x * )) 2 + τ 2 , thus, it follows that m i=1 (λ * i g i (x * ) + τ ) 2 ≤ m i=1 (λ * i g i (x * )) 2 + τ 2 , follows that m i=1 (λ * i g i (x * )) 2 + τ 2 + 2τ λ * i g i (x * ) ≤ m i=1 (λ * i g i (x * )) 2 + τ 2 , so, m i=1 λ * i g i (x * ) ≤ 0,
and since λ * i ≥ 0 and g i (x * ) ≥ 0, i = 1, ..., m, it must be true λ * i g i (x * ) = 0, i = 1, ..., m. From denition of saddle point, we know that L H (x, λ * , τ ) ≥ L H (x * , λ * , τ ), by (4.25) and by denition of L H , we can write On the other hand, once again considering property P2 of HPF, for any feasible point

f (x * ) + mτ = L H (x * , λ * , τ ) ≤ L H (x, λ * , τ ) = f (x) + m i=1 P (g i (x), λ * i , τ ).
x, i.e., g i (x) ≥ 0, i = 1, ...m, we will carry out a work similar to that of (4.16)-(4. [START_REF] Cristofari | An Augmented Lagrangian Method Exploiting an Active-Set Strategy and Second-Order Information[END_REF]), thus, we can obtain

f (x) + m i=1 P (g i (x), λ * i , τ ) ≤ f (x) + mτ, (4.27)
now, we replace (4.27) in (4.26), then follow f (x * ) + τ m ≤ f (x) + τ m, from this last inequality, we obtain f (x * ) ≤ f (x), whenever x is feasible. Therefore, x * is a global optimal solution of (P).

Let's consider the following denitions. Let

F τ (x) = sup λ≥0 L H (x, λ, τ ). Then F τ (x) = f (x) + mτ, if g i (x) ≥ 0, i = 1, ..., m and F τ (x) = ∞, otherwise.
Therefore, we can consider the following problem

x * = argmin {F τ (x) | x ∈ IR n } ,
ϕ τ (λ) ≤ F τ (x) = f (x) + mτ, ∀x ∈ S, ∀ λ ∈ IR m + .
Proof. For any feasible x and λ, we then we can get the weak duality. Indeed, by the denition of ϕ τ , we have

ϕ τ (λ) = inf w∈I R n L H (w, λ, τ ) = inf w∈I R n f (w) + m i=1 P (g i (w), λ i , τ ) ≤ inf w∈S f (w) + m i=1 P (g i (w), λ i , τ ) = f (x) + m i=1 P (g i (x), λ i , τ ). (4.30)
Since we know that x is feasible, we have g i (x) ≥ 0, i = 1, ..., m, immediately then, for the property P2 of the HPF, we get the following expressions (4.32)

P (g i (x), λ i , τ ) ≤ P (0, λ i , τ ), i = 1, ..., m,
If x and λ are feasible solutions of the primal and dual problems and F τ (x) = ϕ τ ( λ), then x = x * and λ = λ * . From Remark 4.0.1, with the smoothness of f (x) and g i (x), i = 1, ..., m, we ensure the smoothness for the dual function ϕ τ (λ).

ϕ τ (λ * ) = min x∈I R n L H (x, λ * , τ ) = L H (x * , λ * , τ ) ≥ L H (x * , λ, τ ) ≥ min x∈I R n L H (x, λ, τ ) = ϕ τ (λ), ∀λ ≥ 0. Therefore ϕ τ (λ * ) = max ϕ τ (λ) | λ ∈ IR m + , in this way λ * ∈ IR m
+ is a solution of the dual problem and since we have L H (x * , λ * , τ ) = f (x * ) + mτ, so (4.33) hold.

Proposition 4.1.3 Suppose that (4.33) holds, for the viable points x * and λ * , then x * is a solution of the problem (P) and λ * is a solution of the dual problem (4.29).

Proof. Let g i (x * ) ≥ 0, i = 1, ..., m, with x * ∈ S, λ * i ≥ 0, i = 1, ..., m and (4.33) with τ > 0 xed. Then for (4.32) where x and λ are viable, we can obtain the following

f (x) + mτ ≥ ϕ τ (λ * ) = f (x * ) + mτ ≥ ϕ τ (λ),
that is, x * is solution of the problem (P) and λ * is solution of (4.29), which corresponds

Test Problems

With the following examples proposed in the book [START_REF] Hock | Test Examples for Nonlinear Programming Codes[END_REF], we are going to verify the strong duality. On the other hand, in each example, the value of m means the total number of restrictions. Also, in all the examples starting points are considered, so that assumption C2 is veried.

min x∈I R 2 f (x) = 100(x 2 -x 2 1 ) 2 + (1 -x 1 ) 2 s.t. g 1 (x) = x 2 + 1.5 ≥ 0.
Starting with x 0 = (-2, 1) (feasible), f (x 0 ) = 909 and m = 1. The minimum value is f (x * ) = 0 at the optimal solution x * = (1, 1).

min x∈I R 3 f (x) = x 2 1 + x 2 2 + x 2 3 s.t. g 1 (x) = x 2 1 + x 2 2 -1 ≥ 0, g 2 (x) = x 1 -1 ≥ 0, g 3 (x) = 10 -x 1 ≥ 0, g 4 (x) = x 2 + 10 ≥ 0, g 5 (x) = 10 -x 2 ≥ 0, g 6 (x) = x 3 + 10 ≥ 0, g 7 (x) = 10 -x 3 ≥ 0.
Starting with x 0 = (1, 1, 1) (feasible), f (x 0 ) = 3 and m = 7. The minimum value is f (x * ) = 1 at the optimal solution x * = (1, 0, 0).

min x∈I R 3 f (x) = 0.2x 3 -0.8x 1 s.t. g 1 (x) = x 2 -e x 1 ≥ 0, g 2 (x) = x 3 -e x 2 ≥ 0, g 3 (x) = x 1 ≥ 0, g 4 (x) = x 2 ≥ 0, g 5 (x) = x 3 ≥ 0, g 6 (x) = 100 -x 1 ≥ 0, g 7 (x) = 100 -x 2 ≥ 0, g 8 (x) = 10 -x 3 ≥ 0.
Starting with x 0 = (0, 1.05, 2.9) (feasible), f (x 0 ) = 0.58 and m = 8. The minimum value is f (x * ) = 0.5181632741 at the optimal solution x * = (0.1841264879, 1.202167873, 3.327322322).

min x∈I R 4 f (x) = x 2 1 + 0.5x 2 2 + x 2 3 + 0.5x 2 4 -x 1 x 3 + x 3 x 4 -x 1 -3x 2 + x 3 -x 4 s.t. g 1 (x) = 5 -x 1 -2x 2 -x 3 -x 4 ≥ 0, g 2 (x) = 4 -3x 1 -x 2 -2x 3 + x 4 ≥ 0, g 3 (x) = x 2 + 4x 3 -1.5 ≥ 0, g 4 (x) = x 1 ≥ 0, g 5 (x) = x 2 ≥ 0, g 6 (x) = x 3 ≥ 0, g 7 (x) = x 4 ≥ 0.
Starting with x 0 = (0.5, 0.5, 0.5, 0.5) (feasible), f (x 0 ) = -1.25 and m = 7. The minimum value is f (x * ) = -4.681818181 at the optimal solution x * = (0.2727273, 2.090909, -0.26E -10, 0.5454545). • The HPF belongs to class C ∞ . Hence, L H (x, λ, τ ) will be class C ∞ if the involved functions f (x) and g i (x), i = 1, ..., m, are too. This is an outstanding property from the computational point of view.

• The smooth behavior of the modied objective function oers the possibility to use the best unconstrained minimization techniques, which use second-order derivatives.

Chapter 6

Future Work

• Although important theoretical points have been developed, we are far from having exhausted our studies. In fact, the connections between hyperbolic penalty and the Lagrangian function extend even further the horizons of new theoretical lines and practical experimentation to be researched.

• Considering this rst work that contains results of existence and convergence with strong assumptions, we consider it a future and natural work to carry out research considering more relaxed assumptions, to obtain more general results.

• Extend the convergence result of the HALA for the nonconvex problem:

min x∈I R n f (x) s.t. h(x) = 0, l ≤ x ≤ u
where f : IR n → IR and h : IR n → IR m are continuously dierentiable functions, l and u are vectors in IR n corresponding to lower and upper bounds in the variable, respectively, see [START_REF] Granville | Optimal reactive dispatch through interior point methods[END_REF]. This model is also studied in [START_REF] Andreani | Augmented Lagrangian methods under the constant positive linear dependence constraint qualication[END_REF], [START_REF] Leyer | An augmented Lagrangian lter method[END_REF], [START_REF] Cristofari | An Augmented Lagrangian Method Exploiting an Active-Set Strategy and Second-Order Information[END_REF] and [START_REF] Torrealba | Augmented Lagrangian algorithms for solving the continuous nonlinear resource allocation problem[END_REF].

We are also interested in solving the following problem min f (x) s.t. g i (x) = 0, i = 1, ..., m < n h i (x) ≤ 0, j = 1, ..., r

x min ≤ x ≤ x max , where x ∈ IR n , g e h are continuously dierentiable, see [START_REF] Baptista | Logarithmic barrier-augmented Lagrangian function to the optimal power ow problem[END_REF]. This model is also studied in [START_REF] Birgin | Improving ultimate convergence of an augmented Lagrangian method[END_REF] and [4].

• Solve the nonlinear programmin problem with equality constraints considering the hyperbolic proximal algorithm

x k+1 ∈ argmin L H (x, λ k , τ ) + 1 2c k x -x k 2 , c k > 0 and update λ k+1 as in (3.3). A similar idea can be seen at [START_REF] Iusem | Augmented Lagrangian methods and proximal point methods for convex optimization[END_REF], [START_REF] Zhang | The rate of convergence of proximal method of multipliers for equality constrained optimization problems[END_REF] and [START_REF] Izmailov | Perturbed Augmented LagrangianMethod Framework with Applications to Proximal and Smoothed Variants[END_REF].

Figure 2 . 1 : 1 Remark 2 . 1 . 1

 211211 Figure 2.1: Hyperbolic Penalty

Lemma 3 . 2 . 1

 321 Let d > 0 and a sequence {a k } ⊂ IR + . If lim k→∞ a k / a k + d = 0 then lim k→∞ a k = 0.

  1), of (3.25) and (3.23), it follows that lim k→∞ λ k+1 i g i (x k+1 ) = 0, i = 1, ..., m.

Theorem 3 . 3 . 1

 331 The convex problem (P) satises C1, C2, C3 and C4. Let sequences {x k } and {λ k } generated by HALA. Then any limit point of a subsequence {x k } and {λ k } are an optimal solution-Lagrange multiplier pair for the problem (P). Proof. Let be τ > 0 xed. By C3 follows the boundedness of the sequence {x k }, and also we know that the sequence {λ k } of the Lagrange multipliers generated by the HALA is bounded, see Proposition 3.2.3. Henceforth, we can consider the following convergent subsequences lim k→∞ x k = x and lim k→∞ λ k = λ with k ∈ K 1 ⊂ IN . Now by C4, we have lim k→∞ g i (x k ) = g i (x) ≥ 0, i = 1, ..., m. From Proposition 3.2.1 we obtain, lim k→∞ λ k i = λi ≥ 0, i = 1, ..., m.(3.27) Passsing the limit in (3.20), we have lim k→∞ λ k i g i (x k ) = λi g i (x) = 0, ∀i = 1, ..., m.

Examples 3 .

 3 4.1 and 3.4.5 are proposed by us. In Tables 3.1-3.11, we can see the feasible starting points considered.

Proposition 4 . 1 . 1

 411 Consider the convex problem (P). Assume the assumption C2 it hold. Then x * ∈ S is a solution of problem (P) for any τ > 0 if and only if:

  L H (x * , λ * , τ ) = f (x * ) + mτ.

(4. 24 )

 24 By(4.24) and denition of L H , we obtain L H (x * , λ * , τ ) = f (x * ) + mτ.

(4. 28 )

 28 that is the problem (P) reduces to solving (4.28).Letϕ τ (λ) = inf x∈I R n L H (x, λ, τ )(possibly ϕ τ (λ) = -∞ for some λ) and consider the following dual problem of (P), that consisting of nding λ * = argmax {ϕ τ (λ) | λ ≥ 0} .

(4. 29 )

 29 In the following result, we are going to verify the weak duality. Proposition 4.1.2 Let x be a feasible solution to problem (P) and let λ be a feasible solution to problem (4.29). Then

we rewrite the expression above, as follows m i=1 PPP

 i=1 (g i (x), λ i , τ ) ≤ m i=1 (0, λ i , τ ), now, we apply property P3, on the right side of the previous inequalitym i=1 P (g i (x), λ i , τ ) ≤ m i=1 τ = mτ,we add f (x), to both sides of the inequality abovef (x) + m i=1 (g i (x), λ i , τ ) ≤ f (x) + mτ,(4.31) we replace (4.31) in (4.30), so ϕ τ (λ) ≤ f (x) + mτ, ∀x ∈ S, ∀λ ∈ IR m + .

Theorem 4 . 1 . 1

 411 The problem (P) is considered. The assumption C2 is veried. Then the existence of a solution of problem (P) implies that the problem (4.29) has a solution and ϕ τ (λ * ) = f (x * ) + mτ, f or any τ > 0.

( 4 . 33 )

 433 Proof. Let x * be a solution of problem (P). By C2, we get λ * ≥ 0, such that (4.5) is veried. So we have

Example 4 . 2 . 1

 421 Problem 1 (HS1).

Example 4 . 2 . 2

 422 Problem 30 (HS30).

Example 4 . 2 . 3

 423 Problem 66 (HS66).

Example 4 . 2 . 4

 424 Problem 76 (HS76).

  Let {λ k } be a sequence generated by HALA. The sequence {Φ(λ k )} is monotone nondecreasing for all k ∈ IN .

	Remark 3.2.2 From inequality (3.9), we can see that iteration (3.3) has the following
	characteristic
	0 < λ k+1 i	< 2λ k i , i = 1, ..., m.	(3.10)
	Remark 3.2.3 From C3 and Proposition 3.2.1, we obtain that HALA is well dened.
	Theorem 3.2.1

  Theorem 3.2.2 Let the sequences {x k } and {λ k } be generated by HALA. Then

	lim k→∞	λ k i g

i (x k ) = 0, i = 1, ..., m.

(3.20) Proof. Let be τ > 0 xed. Since Φ(•) is concave we have the expression (3.15).

We are going to verify that the series in (3.15) is convergent;

(3.15)

gives by summation

Table 3 .

 3 1: Example 3.4.1 with τ = 0.10E -04

	k	x 1	x 2		x 3		f (x)	L H (x, λ, τ ) via
	0	1.5000000000	1.5000000000	1.0000000000	12.2500000000	12.2500000	0
	1	0.1538461538	0.3870967742	0.1525423729	0.5201381914	2.9858922	1
	2	0.2962962963	0.6486486486	0.2647058823	1.5078874026	5.0869781	1
	3	0.5517241379	0.9795918367	0.4186046511	3.6628294023	7.8299588	1
	4	0.9696969696	1.3150684931	0.5901639344	7.4171333201	10.4781607	1
	5	1.3958664938	1.5256298991	0.7069243209	11.3792683000	11.3793010	1
	6	1.3958680939	1.5256305407	0.7069246891	11.3792836271	11.3793036	0
	7	1.3958680939	1.5256305407	0.7069246891	11.3792836271	11.3793036	0
			Table 3.2: Example 3.4.1 with τ = 0.10E -04
				g 1 (x)		g 2 (x)
			k	via	λ 1	via	λ 2
			0	0	0.0100000	0	0.0100000
			1	0	0.0000000	1	0.0200000
			2	0	0.0000000	1	0.0400000
			3	0	0.0000000	1	0.0800000
			4	0	0.0000000	1	0.1600000
			5	0	0.0000000	1	0.2680100
			6	0	0.0000000	0	0.2680103
			7	0	0.0000000	0	0.2680107

Table 3 .

 3 3: Example 3.4.2 with τ = 0.10E -05

	k	x 1		x 2		x 3		x 4	f (x)	L H (x, λ, τ ) via
	0	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
	1	2.3074726	2.3204238	5.0921913	-3.2752419	-79.7053622	-77.0243381
	2	2.2312597	2.2670664	4.9403997	-3.0011307	-79.3079453	-74.4427220
	3	1.9829192	2.0720262	4.6755395	-2.7161503	-78.1500359	-69.9582510
	4	1.6153987	1.7745712	4.2248525	-2.1925613	-74.7549829	-63.0318244
	5	1.0327825	1.2993558	3.5507600	-1.4755845	-66.4076345	-54.3478952
	6	0.5301862	0.8883313	2.5995884	-0.8980092	-52.5776378	-47.4647475
	7	0.1198347	1.0617978	2.1622807	-0.9736842	-46.6910951	-44.2047663
	8	0.0000002	1.0000001	2.0000001	-0.9999997	-44.0000009	-43.9999972
	9	0.0000000	1.0000000	2.0000000	-1.0000000	-44.0000000	-43.9999970
	10	0.0000000	1.0000000	2.0000000	-1.0000000	-44.0000000	-43.9999970
				Table 3.4: Example 3.4.2 with τ = 0.10E -05
					g 1 (x)		g 2 (x)	g 3 (x)
			k	via		λ 1	via	λ 2	via	λ 3
			0	0	0.0100000	0	0.0100000	0	0.0100000
			1	1	0.0200000	1	0.0200000	1	0.0200000
			2	1	0.0400000	1	0.0400000	1	0.0400000
			3	1	0.0800000	1	0.0800000	1	0.0800000
			4	1	0.1600000	1	0.1600000	1	0.1600000
			5	1	0.3200000	1	0.3200000	1	0.3200000
			6	1	0.6400000	1	0.6400000	1	0.6400000
			7	1	1.2800000	0	0.0000000	1	1.2800000
			8	0	1.0000009	0	0.0000000	1	1.9999986
			9	0	1.0000000	0	0.0000000	0	2.0000000
			10	0	0.9999991	0	0.0000000	0	2.0000014
				Table 3.5: Example 3.4.3 with τ = 0.10E -01
	k	x 1				x 2		f (x)	L H (x, λ, τ )	via
	0	0.100000000E+01	0.200000000E+01	-0.500000000E+01	-0.499995000E+01	0
	1	0.166667546E+01	0.999992090E+00	-0.128889633E+02	-0.933330521E+01	1
	2	0.123552039E+01	0.152343892E+01	-0.850782714E+01	-0.848994743E+01	1
	3	0.123477347E+01	0.152466288E+01	-0.849847226E+01	-0.848846423E+01	1
	4	0.123477247E+01	0.152466328E+01	-0.849846350E+01	-0.848846422E+01	0
	5	0.123477247E+01	0.152466328E+01	-0.849846350E+01	-0.848846422E+01	0

Table 3 .

 3

	via

7: Example 3.4.4 with τ = 0.10E + 01 k x f (x) L H (x, λ, τ )

Table 3 .

 3 9: Example 3.4.5 with τ = 0.10E + 00

Table 3 .

 3 11: Example 3.4.5, continuation of Table3.10 with τ = 0.10E + 00

	9 (x)
	g
	g 8 (x)
	g 7 (x)
	g 6 (x)

Table 3 .

 3 12: Example 3.4.6 with τ = 0.10E + 01

	k	x		f (x)	L H (x, λ, τ )	via
	0	0.100000000E+01	0.000000000E+00	0.414213562E+00	0
	1	0.454823513E+08	0.000000000E+00	0.745058060E-08	0
	2	0.536830181E+24	0.000000000E+00	0.745058060E-08	0
	3	0.536830181E+24	0.000000000E+00	0.999999993E+00	0
		Table 3.13: Example 3.4.6 with τ = 0.10E + 01
				g 1 (x)
		k	via	λ 1
		0	0	0.100000000E+01
		1	0	0.111022302E-15
		2	0	0.123259516E-31
		3	0	0.123259516E-31
		Table 3.14: Example 3.4.7 with τ = 0.10E + 01
	k	x		f (x)	L H (x, λ, τ )	via
	0	-0.200000000E+01	0.100000000E+01	0.145732056E+01	0
	1	-0.374299419E+02	0.100000000E+01	0.141421356E+01	0
	2	-0.374299419E+02	0.100000000E+01	0.174911755E+01	0
		Table 3.15: Example 3.4.7 with τ = 0.10E + 01
				g 1 (x)
		k	via	λ 1
		0	0	0.100000000E+01
		1	0	0.292893219E+00
		2	0	0.210565435E+00

Table 4 .

 4 * is reported in the Table4.12.In Table4.14: we can see that HALA is more ecient in the sense that it uses fewer iterations with respect to the other algorithms. We can observe in the computational results that the HALA remains in the viable region in all the examples. On the other hand, despite being the theory developed in this work on convexity hypothesis, our algorithm shows in the Example 4.2.1 that it can also solve non-convex problems.

	1: Example 4.2.1

Table 4 .

 4 8: Continuation of Table 4.7

				g 5 (x)		g 6 (x)			g 7 (x)	g 8 (x)
		k	via		λ 5	via	λ 6		via	λ 7	via	λ 8
		0	0	0.100000000E+02	0	0.100000000E+02	0	0.100000000E+02	0	0.100000000E+02
		1	0	0.450750548E-12	0	0.111022302E-14	0	0.000000000E+00	0	0.113242749E-12
		2	0	0.450750480E-12	0	0.111022301E-14	0	0.000000000E+00	0	0.113242740E-12
		3	0	0.450750413E-12	0	0.111022300E-14	0	0.000000000E+00	0	0.113242731E-12
						Table 4.9: Example 4.2.4
	k		x1		x2		x3		x4	f (x)	LH(x, λ, τ )	via
		0.500000000E+00	0.500000000E+00	0.500000000E+00	0.500000000E+00	-0.125000000E+01	-0.125000000E+01	0
		0.272727650E+00	0.209090766E+01	0.147253122E-05	0.545452356E+00	-0.468181418E+01	-0.468180958E+01	0
		0.272727273E+00	0.209090909E+01	0.413220517E-10	0.545454545E+00	-0.468181818E+01	-0.468174818E+01	0
						Table 4.10: Example 4.2.4
				g 1 (x)			g 2 (x)			g 3 (x)	g 4 (x)
	k	via		λ 1	via	λ 2		via	λ 3	via	λ 4
	0	0	0.100000000E+02	0	0.100000000E+02	0	0.100000000E+02	0	0.100000000E+02
	1	0	0.454545522E+00	0	0.186739513E-11	0	0.143196566E-10	0	0.672217837E-10
	2	0	0.454545455E+00	0	0.186739456E-11	0	0.143196445E-10	0	0.672216605E-10

Table 4 .

 4 11: Continuation of Table 4.10

			g 5 (x)		g 6 (x)		g 7 (x)
	k	via	λ 5	via	λ 6	via	λ 7
	0	0	0.100000000E+02	0	0.100000000E+02	0	0.100000000E+02
	1	0	0.114352972E-11	0	0.172728506E+01	0	0.168043357E-10
	2	0	0.114352972E-11	0	0.172728506E+01	0	0.168043357E-10

Table 4 .

 4 13: Example 4.2.5

k x 1

Table 4 .

 4

					14: Iterations
	N	HALA	Alg1	Alg2	Alg3	Alg4	Alg5	Alg6	Alg7	Alg8	Alg9
	HS1	2	18	34	32	40	24	36	260	36	27
	HS30	2	3	8	11	7	7	10	7	11	10
	HS66	3	12	12	13		11	11	23	5	20
	HS76	2	28	9	11	10		9	23	7	12
	HS100	2	18	10	11	15	9	14	99	13	14

•

  The results presented in this work provide the necessary theoretical framework for the construction of a new algorithm to which we give the name Hyperbolic Augmented Lagrangian Algorithm. The convergence of the algorithm proposed was also demonstrated. In this way, we introduce a new algorithm in the area of mathematical optimization.

the validity of the strong duality.

Computational Illustration

We use HALA to guarantee the theory proposed in this work. The program were compiled by the GNU Fortran compiler version 4:7.4.0-1ubuntu2. [START_REF] Andreani | Augmented Lagrangian methods under the constant positive linear dependence constraint qualication[END_REF].

The numerical

Experiments are conducted on a Notebook with operating system Ubuntu 18.04.5, CPU i7-3632QM and 8GB RAM. The unconstrained minimization tasks were carried out by means of a Quasi-Newton algorithm employing the BFGS updating formula, with the function VA13 from HSL library [START_REF]Harwell Subroutine Library: A collection of Fortran codes for large scale scientic computation[END_REF]. The algorithm stop when the solution is viable (feasible) an the absolute value of the dierence of the two consecutive solutions

x k -x k-1 is less than 1.D -5.

We are going to take advantage of this section to make some comparisons of our algorithm HALA (see Table 4.14) with respect to the following algorithms:

Alg1= [START_REF] Di Pillo | A truncated Newton method for constrained optimization[END_REF] which is an truncated Newton method;

Alg2= [START_REF] Griva | Global convergence of a primal-dual interior point method for nonlinear programming[END_REF] which is a primal-dual interior point method;

Alg3= [START_REF] Vanderbei | Interior-point algorithm for nonconvex nonlinear programming[END_REF] which is an interior-point algorithm;

Alg4= [START_REF] Qi | A new qp-free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization[END_REF] which is a QP-free method;

Alg5= [START_REF] Bakhtiari | A simple primal-dual feasible interior-point method for nonlinear programming with monotone descent[END_REF] which is a primal-dual feasible interior-point method;

Alg6= [START_REF] Yang | A feasible sequential linear equation method for inequality constrained optimization[END_REF] which is a feasible sequential linear equation algorithm;

Alg7= [START_REF] Wang | An inexact rst-order methodd for constrained nonlinear optimization[END_REF] which is an inexact rst-order method;

Alg8= [START_REF] Herskovits | Feasible direction interior-point technique for nonlinear optimization[END_REF] which is a feasible direction interior-point technique;

Alg9= [START_REF] Akrotirianakis | Globally convergent interior-point algorithm for nonlinear programming[END_REF] which is an interior point algorithm.

Example 4.2.5 Problem 100 (HS100).

Starting with x 0 = (1, 2, 0, 4, 0, 1, 1) (feasible), f (x 0 ) = 714 and m = 4.

The minimum value is f (x * ) = 680.6300573 at the optimal solution x * = (2.330499, 1.951372, -0.4775414, 4.365726, -0.6244870, 1.038131, 1.594227).

Results

For each table, the letter N indicates the name of the problem, λ is the multiplier Lagrange, x is the primal variable, f (x) is the value of the objective function, g i (x) are the constraints of each problem, L H (•, •, •) is the value of the HALF and via = viable = f easible where, in each iteration, the obtained point can be viable, then its value is "0 = yes ′′ or the point can be inviable, then the value is "1 = not ′′ and τ is the penalty parameter. In all of our examples, we will use τ = 0.10E -04. We are going to analyze the Examples.

• Example 4.2.1: The HALA solves this example even though function f is nonconvex, see Tables 4.1 and 4.2.

• Example 4.2.2: the function f is strictly convex. From Table 4.3, we can see that in iteration 2 the Theorem 4.1.1 can be veried, that is, we have the following f (x * ) + mτ = 1.00000000 + (7)(0.00001) = 1.00007 and ϕ τ (λ * ) = L H (x * , λ * , τ ) = 1.00007, then, ϕ τ (λ * ) = f (x * ) + mτ. So, x * = (0.100000000E + 01, 0.100000000E + 01) is the solution of the primal problem and from Table 4.4 and Table 4.5, we can see the λ * is the solution of the dual problem in the iteration 2.

• Example 4.2.3: the function f is linear. From Table 4.6, we can see that in iteration 3 the Theorem 4.1.1 can be veried, that is, we have the following f (x * ) + mτ = 0.518163274 + (8)(0.00001) = 0.518243274 and ϕ τ (λ * ) = L H (x * , λ * , τ ) = 0.518243274, then, ϕ τ (λ * ) = f (x * ) + mτ. So, x * is the solution of the primal problem and from Table 4.7 and Table 4.8, we can see the λ * is the solution of the dual problem in the iteration 3.

• Example 4.2.4: the function f is strictly convex. From Table 4.9, we can see that in iteration 2, the Theorem 4.1.1 can be veried, that is, we have the following f (x * ) + mτ = -4.68181818 + (7)(0.00001) = -4.68174818