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Bilevel optimization problems involve two nested objectives, where anupper-level objective depends on a solution to a lower-level problem. When thelatter is non-convex, multiple critical points may be present, leading to anambiguous definition of the problem. In this paper, we introduce a keyingredient for resolving this ambiguity through the concept of a selection mapwhich allows one to choose a particular solution to the lower-level problem.Using such maps, we define a class of hierarchical games between two agentsthat resolve the ambiguity in bilevel problems. This new class of gamesrequires introducing new analytical tools in Morse theory to extend implicitdifferentiation, a technique used in bilevel optimization resulting from theimplicit function theorem. In particular, we establish the validity of such amethod even when the latter theorem is inapplicable due to degenerate criticalpoints. Finally, we show that algorithms for solving bilevel problems based onunrolled optimization solve these games up to approximation errors due tofinite computational power. A simple correction to these algorithms is thenproposed for removing these errors.

Non-Convex Bilevel Games with Critical Point Selection Maps