Bilevel Programming Problems: A view through Set-valued Optimization

TL;DR

This paper investigates the general case of bilevel programming problems using set-valued optimization, analyzing solution concepts and comparing its advantages to traditional optimistic and pessimistic formulations.

Contribution

It extends the set-valued formulation of bilevel problems to the general case, beyond the continuous set-up with value attainment assumptions, and explores the relationships among different solution notions.

Findings

Set-valued formulation may not offer significant advantages over existing formulations.

The study clarifies the connection among various solution concepts in bilevel programming.

The general case of set-valued formulation is analyzed beyond previous assumptions.

Abstract

Bilevel programming is one of the very active areas of research with many real-life applications in economics and engineering. Bilevel problems are hierarchical problems consisting of lower-level and upper-level problems, respectively. The leader or the decision-maker for the upper-level problem decides first, and then the follower or the lower-level decision-maker chooses his/her strategy. In the case of multiple lower-level solutions, the bilevel problems are not well defined, and there are many ways to handle such a situation. One standard way is to put restrictions on the lower level problems (like strict convexity) so that nonuniqueness does not arise. However, those restrictions are not viable in many situations. Therefore, there are two standard formulations, called pessimistic formulations and optimistic formulations of the upper-level problem. A set-valued formulation has been proposed and has been studied in the literature. However, the study is limited to the continuous set-up with the assumption of value attainment, and the general case has not been considered. In this paper, we focus on the general case and study the connection among various notions of solution. Our main findings suggest that the set-valued formulation may not hold any bigger advantage than the existing optimistic and pessimistic formulation.