NonConvex Bilevel Games with Critical Point Selection Maps
Abstract
Bilevel optimization problems involve two nested objectives, where an upperlevel objective depends on a solution to a lowerlevel problem. When the latter is nonconvex, multiple critical points may be present, leading to an ambiguous definition of the problem. In this paper, we introduce a key ingredient for resolving this ambiguity through the concept of a selection map which allows one to choose a particular solution to the lowerlevel problem. Using such maps, we define a class of hierarchical games between two agents that resolve the ambiguity in bilevel problems. This new class of games requires introducing new analytical tools in Morse theory to extend implicit differentiation, a technique used in bilevel optimization resulting from the implicit function theorem. In particular, we establish the validity of such a method even when the latter theorem is inapplicable due to degenerate critical points. Finally, we show that algorithms for solving bilevel problems based on unrolled optimization solve these games up to approximation errors due to finite computational power. A simple correction to these algorithms is then proposed for removing these errors.
 Publication:

arXiv eprints
 Pub Date:
 July 2022
 DOI:
 10.48550/arXiv.2207.04888
 arXiv:
 arXiv:2207.04888
 Bibcode:
 2022arXiv220704888A
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Differential Geometry