On the Nature of Regularity Assumptions in Bilevel Optimization with Constrained Lower-level Problem

In this paper, we study the regularity assumptions commonly adopted in bilevel optimization with constrained lower-level problems, including the linear independence constraint qualification, the strict complementary slackness condition, and the second-order sufficient condition. These conditions are typically required to hold for the lower-level problem at every upper-level variable $x$. We first show that the requirement that these conditions hold at every upper-level variable $x$ is strong, in the sense that it is non-prevalent: there exist problems for which no sufficiently small perturbation of the lower-level objective and constraints can make the conditions hold at every $x$. To establish the result, we prove rigidity theorems showing that certain structural quantities of the lower-level problem must remain invariant across all $x$ whenever these conditions hold everywhere. We then construct explicit counterexamples in which these invariants differ between two values of $x$. In contrast, we show that the weaker requirement, that these conditions hold at almost every $x$, is a weak assumption, in the sense that it is prevalent: with probability one over a random perturbation of the lower-level objective and constraints, each condition holds at almost every $x$. We further analyze the gap between the two requirements. Although the ``every $x$'' and ``almost every $x$'' versions differ only on a measure-zero set, we show that this difference introduces fundamental difficulties in both theory and computation for bilevel optimization.