This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

Well, someone can decide by themselves what they want to do and need to do but sometimes, that kind of person will need some theory of vector optimization references. People with open minded will always try to seek for the new things and information from many sources. On the contrary, people with closed mind will always think that they can do it by their principals. So, what kind of person are you?

Theory Of Vector Optimization

In this paper, we consider three kinds of pointwise well-posedness for set optimization problems. We establish some relations among the three kinds of pointwise well-posedness. By virtue of a generalized nonlinear scalarization function, we obtain the equivalence relations between the three kinds of pointwise well-posedness for set optimization problems and the well-posedness of three kinds of scalar optimization problems, respectively.

Scalarization andpointwise well-posedness for set optimization problems

In this paper, we consider the vector equilibrium problems involving lexicographic cone in Banach spaces. We introduce the new concepts of the Tykhonov well-posedness for such problems. The corresponding concepts of the Tykhonov well-posedness in the generalized sense are also proposed and studied. Some metric characterizations of well-posedness for such problems are given. As an application of the main results, several results on well-posedness for the class of lexicographic variational inequalities are derived.

Tykhonov well-posedness for lexicographic equilibrium problems

In this paper set optimization problems with three types of set order relations are concerned. We introduce various types of Levitin–Polyak (LP) well-posedness for set optimization problems and survey their relationships. After that, sufficient and necessary conditions for the reference problems to be LP well-posed are given. Furthermore, using the Kuratowski measure of noncompactness, we study characterizations of well-posedness for set optimization problems. Moreover, the links between stability and LP well-posedness of such problems are established via the study on approximating solution mappings. Tools and techniques used in this study and our results are different from existing ones in the literature.

Levitin–Polyak well-posedness for set optimization problems involving set order relations

This article investigates stability conditions for set optimization problems with the set less order relation in the senses of Panileve–Kuratowski and Hausdorff convergence. Properties of various kinds of convergences for elements in the image space are discussed. Taking such properties into account, formulations of internal and external stability of the solutions are studied in the image space in terms of the convergence of a solution sets sequence of perturbed set optimization problems to a solution set of the given problem.

Convergence of Solutions to Set Optimization Problems with the Set Less Order Relation

We consider lexicographic vector equilibrium problems in metric spaces. Sufficient conditions for a family of such problems to be (uniquely) well posed at the reference point are established. As an application, we derive several results on well-posedness for a class of variational inequalities.

http://www.optimization-online.org/DB_FILE//2013/01/3740.pdf

Well-Posedness for Lexicographic Vector Equilibrium Problems

Inspired by the great importance of equilibrium problems and the lexicographic order, we consider a parametric lexicographic equilibrium problem. Sufficient conditions for the upper semicontinuity, closedness, and continuity of solution maps are established. Many examples are provided to ensure the essentialness of the imposed assumptions. Applications to lexicographic variational inequalities and lexicographic optimization problems are discussed.

Continuity properties of solution maps of parametric lexicographic equilibrium problems

In this article, we consider vector optimization problems with uncertain data. We first formulate optimistic counterparts of the reference problems and propose concepts of efficient solutions to such counterparts. We then introduce concepts of pointwise and global well-posedness for optimistic counterparts. Using the generalized Gerstewitz’s function and properties of elements in the image space, we establish the relationships between well-posedness properties for the reference problems and that for scalar optimization ones. Based on such relations, we have studied sufficient conditions of these well-posedness properties for the considered problems via the corresponding scalar problems. Finally, by virtue of a forcing function, the characterizations of the two concepts of well-posedness for such problems are presented.

Tran Quoc Duy

Papers

Convergence of Solutions to Set Optimization Problems with the Set Less Order Relation

Tykhonov well-posedness for lexicographic equilibrium problems

Well-Posedness for Lexicographic Vector Equilibrium Problems

Continuity properties of solution maps of parametric lexicographic equilibrium problems

Well-posedness for the optimistic counterpart of uncertain vector optimization problems