🤖 AI Summary
Bilevel programming lacks a well-defined solution concept when multiple lower-level optimal solutions exist; existing optimistic/pessimistic formulations rely on strong assumptions—such as strict convexity or value attainability—that limit applicability.
Method: Breaking free from continuity and value-attainability constraints, we systematically investigate the logical relationships and equivalence conditions among optimistic, pessimistic, and set-valued solution concepts within a set-valued optimization framework. Using tools from set-valued analysis, nonsmooth optimization, and variational geometry, we construct a unified comparative framework.
Contribution/Results: We rigorously characterize the inclusiveness and limitations of these three solution paradigms. A key finding is that the set-valued formulation does not inherently outperform classical optimistic/pessimistic models in general settings; its theoretical advantages require additional structural assumptions. This work establishes rigorous criteria for selecting appropriate solution concepts in bilevel modeling, offering both theoretical foundations and practical guidance.
📝 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.