“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.

https://www.heldermann-verlag.de/jca/jca11/jca0391.pdf

Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings

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

Minmax Robustness for Multi-objective Optimization Problems

Nonconvex scalarization in set optimization with set-valued maps

On cone of convexity of set-valued maps

On set-valued optimization

In optimization with set-valued maps, there are two types of criteria of solutions; one is vector optimization, which is the usual one, and the other is set optimization, which is based on a natural idea and defined by this paper. The aim of the paper is to study set optimization. We consider criteria of solutions of set optimization, show some examples with respect to the criteria, introduce some type of semicontinuities for set-valued maps, and show existence theorems of solutions.

Existence theorems of set optimization with set-valued maps

In this paper, we discuss the connection between concepts of robustness for multi-objective optimization problems and set order relations. We extend some of the existing concepts to general spaces and cones using set relations. Furthermore, we derive new concepts of robustness for multi-objective optimization problems. We point out that robust multi-objective optimization can be interpreted as an application of set-valued optimization. Furthermore, we develop new algorithms for solving uncertain multi-objective optimization problems. These algorithms can be used in order to solve a special class of set-valued optimization problems.

/pdf/the-relationship-between-multi-objective-robustness-concepts-24egzvg9fp.pdf

Daishi Kuroiwa

Papers

On cone of convexity of set-valued maps

On set-valued optimization

Existence theorems of set optimization with set-valued maps

The relationship between multi-objective robustness concepts and set-valued optimization

Convexity for set-valued maps