Stability of the intersection of solution sets of semi-infinite systems

Many mathematical programming models arising in practice present a block structure in their constraint systems. Consequently, the feasibility of these problems depends on whether the intersection of the solution sets of each of those blocks is empty or not. The existence theorems allow to decide when the intersection of non-empty sets in the Euclidean space, which are the solution sets of systems of (possibly infinite) inequalities, is empty or not. In those situations where the data (i.e., the constraints) can be affected by some kind of perturbations, the problem consists of determining whether the relative position of the sets is preserved by sufficiently small perturbations or not. This paper focuses on the stability of the non-empty (empty) intersection of the solutions of some given systems, which can be seen as the images of set-valued mappings. We give sufficient conditions for the stability, and necessary ones as well; in particular we consider (semi-infinite) convex systems and also linear systems. In this last case we discuss the distance to ill-posedness.     

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.     

Continuous selections for set-valued mappings with finite-dimensional convex values

We show selection theorems for set-valued mappings with finite-dimensional convex values, one of which is a partial extension of a selection theorem due to S. Barov (2008) [1].     

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.     

Conic systems and sublinear mappings: equivalent approaches

It is known that linear conic systems are a special case of set-valued sublinear mappings. Hence the latter subsumes the former. In this note we observe that linear conic systems also contain set-valued sublinear mappings as a special case. Consequently, the former also subsumes the latter.     

Generic Existence of Fixed Points for Set-Valued Mappings

We first consider a complete metric space of nonexpansive set-valued mappings acting on a closed convex subset of a Banach space with a nonempty interior, and show that a generic mapping in this space has a fixed point. We then establish analogous results for two complete metric spaces of set-valued mappings with convex graphs.     

Duality for Set-Valued Multiobjective Optimization Problems,Part 1: Mathematical Programming

The duality of multiobjective problems is studied with the help of the apparatus of conjugate set-valued mappings introduced by the author. In this paper (Part 1), a duality theory is developed for set-valued mappings, which is then used to derive dual relations for some general multiobjective optimization problems which include convex programming and optimal control problems. Using this result, in the companion paper (Part 2), duality theorems are proved for multiobjective quasilinear and linear optimal control problems. The theory is applied to get dual relations for some multiobjective optimal control problem.     

Uniform convexity and the splitting problem for selections

We continue to investigate cases when the Repovš–Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We consider the situation in infinite-dimensional uniformly convex Banach spaces. We use the notion of Polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets (not necessary balls). We study general geometric properties of uniformly convex sets. We also obtain an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images.     

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.     

On the stability of closed-convex-valued mappings and the associated boundaries

This paper deals with the stability properties of those set-valued mappings from locally metrizable spaces to Euclidean spaces for which the images are the convex hull of their boundaries (i.e., they are closed convex sets not containing a halfspace). Examples of this class of mappings are the feasible set and the optimal set of convex optimization problems, and the solution set of convex systems, when the data are subject to perturbations. More in detail, we associate with the given set-valued mapping its corresponding boundary mapping and we analyze the transmission of the stability properties (lower and upper semicontinuity, continuity and closedness) from the given mapping to its boundary and vice versa.     

On coderivatives and Lipschitzian properties of the dual pair in optimization

In this paper, we apply the concept of coderivative and other tools from the generalized differentiation theory for set-valued mappings to study the stability of the feasible sets of both the primal and the dual problem in infinite-dimensional linear optimization with infinitely many explicit constraints and an additional conic constraint. After providing some specific duality results for our dual pair, we study the Lipschitz-like property of both mappings and also give bounds for the associated Lipschitz moduli. The situation for the dual shows much more involved than the case of the primal problem.     

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.     

Geometric approach to convex subdifferential calculus

In this paper, we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic results of convex subdifferential calculus in full generality and also derive new results of convex analysis concerning optimal value/marginal functions, normals to inverse images of sets under set-valued mappings, calculus rules for coderivatives of single-valued and set-valued mappings, and calculating coderivatives of solution maps to parameterized generalized equations governed by set-valued mappings with convex graphs.     

On mappings covering at a nonlinear rate and their perturbation stability

The present paper contains a study of covering (alias, openness) properties at a nonlinear rate for set-valued mappings between metric spaces. Such study is focussed on the stability of these properties in the presence of perturbations. A crucial result valid for linear openness, known as Milyutin’s theorem, is extended to set-valued mappings covering at a nonlinear rate under possibly non-Lipschitz perturbations. Consequently, a Lyusternik type theorem is derived from such extension and a general penalization principle for constrained optimization problems, which exploits nonlinear covering properties, is presented.     

Modified continuity and a generalisation of Michael's selection theorem

We modify the definitions of continuity and lower semicontinuity for single-valued mappings and upper and lower semicontinuity for set-valued mappings. For single-valued mappings we have a generalisation of Osgood's theorem and for set-valued mappings we have an extension of Fort's theorem and a generalisation of Michael's selection theorem producing a densely defined selection with a natural continuity property relative to the domain.     

集值映射与不动点

应用D关于β对应关系是正常的构架,引用子集法,通过取凸包与闭包,由单值映射构造集值映射,利用Kakutani不动点定理,讨论几种类型的集值映射的不动点.     

Weak Subdifferential of Set-Valued Mappings

In this note we introduce a notion of the weak contingent generalized gradient for set-valued mappings associated with the contingent epiderivative of set-valued mappings introduced in "E. Bednarczuk and W. Song (1998). Contingent epiderivative and its applications to set-valued optimization. Control and Cybernetics, 27, 376-386; G.Y. Chen and J. Jahn (1998). Optimally conditions for set-valued optimization problems. Mathematical Methods of Operations Research, 48, 187-200." and prove that, under some additional condition, it coincides with the weak subdifferential introduced in "T. Tanino (1992). Conjugate duality in vector optimization. Journal of Mathematical Analysis and Applications, 167, 84-97." when the set-valued map is cone-convex. We also study the weak contingent generalized gradient of a sum of two set-valued mappings and optimality conditions for a set-valued vector optimization problem.     

参数凸二次规划的线性稳定性

本文研究参数凸二次规划的最优解集的稳定性。首先给出参数数学规划的方向线性稳定的定义，然后利用集值映射的微分理论证明线性约束参数凸二次规划是线性稳定的。     

局部凸空间非扩张型集值映射的不动点定理

本文讨论局部凸线性拓补空间非扩张型集值映射的不动点问题，并给出关于单值映射族具有公共不动点的结果。     

Common fixed point theorems and minimax inequalities in locally convex Hausdorff topological vector spaces

In this article, we introduce the concept of a family of set-valued mappings generalized Knaster–Kuratowski–Mazurkiewicz (KKM) w.r.t. other family of set-valued mappings. We then prove that if X is a nonempty compact convex subset of a locally convex Hausdorff topological vector space and 𝒯 and 𝒮 are two families of self set-valued mappings of X such that 𝒮 is generalized KKM w.r.t. 𝒯, under some natural conditions, the set-valued mappings S ∈ 𝒮 have a fixed point. Other common fixed point theorems and minimax inequalities of Ky Fan type are obtained as applications.