A new scheme for approximating the weakly efficient solution set of vector rational optimization problems

TL;DR

This paper introduces a novel polynomial approximation scheme for the weakly efficient solution set in vector rational optimization problems, utilizing semidefinite programming to generate explicit approximations.

Contribution

It presents a new method to approximate the weakly efficient solution set using polynomial and semidefinite programming techniques, which is a significant advancement in vector optimization.

Findings

The scheme effectively approximates the weakly efficient solution set.

The construction reduces to solving semidefinite programming problems.

Numerical examples demonstrate the method's practicality and accuracy.

Abstract

In this paper, we provide a new scheme for approximating the weakly efficient solution set for a class of vector optimization problems with rational objectives over a feasible set defined by finitely many polynomial inequalities. More precisely, we present a procedure to obtain a sequence of explicit approximations of the weakly efficient solution set of the problem in question. Each approximation is the intersection of the sublevel set of a single polynomial and the feasible set. To this end, we make use of the achievement function associated with the considered problem and construct polynomial approximations of it over the feasible set from above. Remarkably, the construction can be converted to semidefinite programming problems. Several nontrivial examples are designed to illustrate the proposed new scheme.