Bilevel Programming Problems: A view through Set-valued Optimization
Kuntal Som, Thirumulanathan D, Joydeep Dutta
TL;DR
This work addresses bilevel programming with potentially non_unique lower_level responses by examining its relationship to set_valued optimization. It develops and leverages the notions of l_minimal and u_minimal solutions within a set_valued framework to connect with real and standard optimistic/pessimistic formulations, highlighting the crucial role of value attainment for equivalence. Using tools such as Berge's Maximum Theorem, the authors delineate when the set_valued formulation aligns with traditional reformulations and when it provides a more natural or complete description, including implications for infinite-dimensional settings. The results offer a coherent, unified view that helps explain when optimistic/pessimistic approaches are effectively subsumed by or complementary to the set_valued upper-level problem, informing theory and potential applications in economics and engineering.
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.