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

Abstract

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.

Set Optimization—A Rather Short Introduction

“…After 2005, many authors adopted the concepts related to C and C , see, among an increasing number of others, [1], [69], [78,77,79], [141,142], [120]. Quite recently, robustness for vector optimization problems has been linked to the two (and other) set relations, see [87], [89], [88].…”

Robust counterparts and robust efficient solutions in vector optimization under uncertainty

“…To overcome this deficiency, the method employed in [5,4,12,13] for multi-objective optimization under uncertainty provides a new insight into vector optimization under uncertainty. Specifically, first the multi-objective optimization problem under uncertainty is treated as a set-valued optimization problem, and then some solution concepts are established on the basis of set order relations, which contain robustness to some extent.…”

“…Specifically, first the multi-objective optimization problem under uncertainty is treated as a set-valued optimization problem, and then some solution concepts are established on the basis of set order relations, which contain robustness to some extent. For more details, see [5,4,12,13] or [11]. In her dissertation [16], Köbis applied this method to uncertain vector optimization and investigated various concepts of robustness systemically.…”

“…Recently much attention has been paid to introducing robustness in multi-objective optimization under uncertainty. For a multi-objective optimization problem with the objective function involving an uncertain parameter, several concepts of robustness were developed without scalarization, such as the componentwise worst case robustness in [17,7], min-max robustness in [5], convex version of min-max robustness in [4] and other ones based on set order relations in [12,13,11]. In this study, we consider the following vector optimization problem…”

Multiple Criteria Decision Making in Finance, Insurance and Investment

“…Then a multiobjective optimization problem, for all realizations of the uncertain parameter, leads to a set optimization problem. Since set optimization problems can also be solved using scalarization techniques (compare, for instance, Hernández and Rodríguez-Marín 2007;Ide et al 2014), our approaches can be adapted to solve such problems as well. Abstract The so-called Problem of Optimal Premium Calculation deals with the selection of the appropriate premiums to be paid by the insurance policies.…”