Openness properties for parametric set-valued mappings and implicit multifunctions

The aim of this paper is to obtain some openness results in terms of normal coderivative for parametric set-valued mappings acting between infinite dimensional spaces. Then, implicit multifunction results are obtained by simply specializing the openness results. Moreover, we study a kind of metric regularity of the implicit multifunction. The results of the paper generalize several recent results in literature.     

Clarke coderivatives of efficient point multifunctions in parametric vector optimization

The paper is devoted to the study of the Clarke/circatangent coderivatives of the efficient point multifunction of parametric vector optimization problems in Banach spaces. We provide inner/outer estimates for evaluating the Clarke/circatangent coderivative of this multifunction in a broad class of conventional vector optimization problems in the presence of geometrical, operator and (finite and infinite) functional constraints. Examples are given for analyzing and illustrating the obtained results.     

Complete characterizations of local weak sharp minima with applications to semi-infinite optimization and complementarity

In this paper, we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials, or tangent cones and subderivatives, or their mixture in finite-dimensional spaces. The results obtained not only extend previous ones in the literature, but also allow us to provide new types of criteria for local weak sharpness. Applications of the developed theory are given to semi-infinite programming and to a new class of semi-infinite complementarity problems.     

Necessary optimality conditions for weak sharp minima in set-valued optimization

The aim of the present paper is to get necessary optimality conditions for a general kind of sharp efficiency for set-valued mappings in infinite dimensional framework. The efficiency is taken with respect to a closed convex cone and as the basis of our conditions we use the Mordukhovich generalized differentiation. We have divided our work into two main parts concerning, on the one hand, the case of a solid ordering cone and, on the other hand, the general case without additional assumptions on the cone. In both situations, we derive some scalarization procedures in order to get the main results in terms of the Mordukhovich coderivative, but in the general case we also carryout a reduction of the sharp efficiency to the classical Pareto efficiency which, in addition with a new calculus rule for Fréchet coderivative of a difference between two maps, allows us to obtain some results in Fréchet form.     

Computation formulas and multiplier rules for graphical derivatives in separable Banach spaces

In this paper, we present new computation formulas for the contingent epiderivative and hypoderivative of a set-valued map taking values in a Banach space with a shrinking Schauder basis. These formulas are established in terms of the Fourier coefficients, and, in particular, in terms of the derivatives of the component maps associated with the Schauder basis. As an application, we obtain multiplier rules for vector optimization problems in terms of the derivatives of the component maps, extending classical results from smooth multiobjective optimization problems.     

Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets

Motivated by the subsmoothness of a closed set introduced by Aussel et al. (2005) [8], we introduce and study the uniform subsmoothness of a collection of infinitely many closed subsets in a Banach space. Under the uniform subsmoothness assumption, we provide an interesting subdifferential formula on distance functions and consider uniform metric regularity for a kind of multifunctions frequently appearing in optimization and variational analysis. Different from the existing works, without the restriction of convexity, we consider several fundamental notions in optimization such as the linear regularity, CHIP, strong CHIP and property (G) for a collection of infinitely many closed sets. We establish relationships among these fundamental notions for an arbitrary collection of uniformly subsmooth closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of closed convex sets to the nonconvex setting.     

Weighted quasi-equilibrium problems with lower and upper bounds

This paper is devoted to studying the solution existence of weighted quasi-equilibrium problems with lower and upper bounds by using maximal element theorems, a fixed point theorem of set-valued mappings and Fan–KKM theorem, respectively. Some new results are obtained.     

On second-order optimality conditions in constrained multiobjective optimization

In this paper we study a multiobjective optimization problem with inequality constraints on finite dimensional spaces. A second-order necessary condition for local weak efficiency is proved under strict differentiability assumptions. We also establish a second-order sufficient condition for local firm efficiency of order 2 under ?-stability assumptions. In this way we generalize some corresponding results obtained by P.Q. Khanh and N.D. Tuan, and by the authors.     

An upper estimate for the Clarke subdifferential of an infimal value function proved via the Mordukhovich subdifferential

The aim of this note is to give an alternative proof for a recent result due to Dorsch et al., which provides an upper estimate for the Clarke subdifferential of an infimal value function. We show the validity of this result under a weaker condition than the one assumed in the aforementioned paper, while the use of the Mordukhovich subdifferential, as an intermediate step, will considerably shorten its proof.     

The Fermat rule for multifunctions for super efficiency

Using variational analysis, we study the vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In terms of the coderivatives and normal cones, we present Fermat’s rules as necessary or sufficient conditions for a super efficient solution of the above problems.     

Generalized invexity of nonsmooth functions

In this paper, the concepts of weak invexity and weak quasi invexity are introduced and the relations among several kinds of generalized invexity are studied for nonsmooth functions by means of the properties of limiting subdifferentials. In addition, the relations between generalized invexity of a nonsmooth function and generalized invariant monotonicity of its limiting subdifferential mapping are researched. Our results here are an extension and generalization of those presented by M. Soleimani-damaneh.     

On the existence of solutions to generalized vector variational-like inequalities

In this paper, we consider a vector version of generalized Minty's lemma and obtain existence theorems of solutions for two kinds of vector variational-like inequalities.     

Openness stability and implicit multifunction theorems: Applications to variational systems

In this paper we aim to present two general results regarding, on one hand, the openness stability of set-valued maps and, on the other hand, the metric regularity behavior of the implicit multifunction related to a generalized variational system. Then, these results are applied in order to obtain, in a natural way, and in a widely studied case, several relations between the metric regularity moduli of the field maps defining the variational system and the solution map. Our approach allows us to complete and extend several very recent results from the literature.     

Scalarization and optimality conditions for vector equilibrium problems

In this paper, we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions, Henig efficient solutions, and globally efficient solutions to the vector equilibrium problems without the convexity assumption. Using nonsmooth analysis and the scalarization results, we provided the necessary conditions for weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems. By the assumption of convexity, we gave sufficient conditions for those solutions. As applications, we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems.     

Necessary optimality conditions for bilevel minimization problems

In this paper we study bilevel minimization problems. Using the implicit function theorem, variational analysis and exact penalty results we establish necessary optimality conditions for these problems.     

Sufficient conditions for pseudo-Lipschitz property in convex semi-infinite vector optimization problems

This paper is devoted to the study of the pseudo-Lipschitz property of the efficient (Pareto) solution map for the perturbed convex semi-infinite vector optimization problem (CSVO). We establish sufficient conditions for the pseudo-Lipschitz property of the efficient solution map of (CSVO) under continuous perturbations of the right-hand side of the constraints and functional perturbations of the objective function. Examples are given to illustrate the obtained results.     

Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.     

Variational sets: Calculus and applications to nonsmooth vector optimization

We develop elements of calculus of variational sets for set-valued mappings, which were recently introduced in Khanh and Tuan (2008) [1] and [2] to replace generalized derivatives in establishing optimality conditions in nonsmooth optimization. Most of the usual calculus rules, from chain and sum rules to rules for unions, intersections, products and other operations on mappings, are established. Direct applications in stability and optimality conditions for various vector optimization problems are provided.