An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces

This paper explores new notions of approximate minimality in set optimization using a set approach. We propose characterizations of several approximate minimal elements of families of sets in real linear spaces by means of general functionals, which can be unified in an inequality approach. As particular cases, we investigate the use of the prominent Tammer–Weidner nonlinear scalarizing functionals, without assuming any topology, in our context. We also derive numerical methods to obtain approximate minimal elements of families of finitely many sets by means of our obtained results

STUDY ON A RELAXATION FOR THEOREMS OF THE ALTERNATIVE FOR SETS (Study on Nonlinear Analysis and Convex Analysis)

In the literature, characterization for set relations are of various application to optimality conditions of set optimization problems, variational principles for set-valued maps, theorems of the alternative, certain robustness of vector optimization problems, and so on. In this paper, the author presents properties of scalarization functions as dual expression of set relations. Comparing to existing results, one can confirm their uniqueness in their relaxed conditions using convex cone-compactness and closedness. Also, we show the results implies generalized Gordan's theorems of the alternative at the last part of the thesis

Robustness of multi-valued optimization problems via set relations (Study on Nonlinear Analysis and Convex Analysis)

This paper focuses on the feasibility of multi-valued optimization problems under perturbations. By using set relations and their scalarization, a modified version of theorems of the alternative can characterize the robustness of the feasibility. Especially under some assumptions, we show algorithms for evaluating the robustness which computers could deal with

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

A steepest descent method for set optimization problems with set-valued mappings of finite cardinality

In this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020)