The Aubin Property for Generalized Equations over $C^2$-cone Reducible Sets

Abstract

This paper establishes the equivalence between the Aubin property and strong regularity for generalized equations over $C^2$-cone reducible sets. This result resolves a long-standing question in variational analysis and extends the classical equivalence theorem for polyhedral sets to a broad class of non-polyhedral sets. Our proof strategy departs from traditional variational techniques, integrating insights from convex geometry with powerful tools from algebraic topology. At the heart of our analysis is a novel index theorem for a class of functions involving metric projections onto arbitrary closed convex sets. Its proof exploits the geometry of normal cones and topological degree theory. We then use a lift of the diffeomorphism provided by the $C^2$-cone reduction to reduce the original generalized equations to the setting of the index theorem. The homological inverse mapping theorem then yields the desired strong regularity. We show that the $C^2$-cone reducibility assumption cannot be removed in general by constructing a three-dimensional semialgebraic counterexample. This result unifies and extends existing stability results for conventional nonlinear programming, nonlinear second-order cone programming, and nonlinear semidefinite programming under a single general framework, and removes the local optimality assumption imposed in the existing non-polyhedral results.

Major revision. Substantial revisions and additions throughout the manuscript