Generic identifiability and second-order sufficiency in tame convex optimization

TL;DR

This paper establishes generic conditions for uniqueness and stability of solutions in tame convex optimization, showing that solutions are identifiable on smooth manifolds and satisfy second-order optimality, ensuring robustness and algorithmic convergence.

Contribution

It proves generic uniqueness, manifold identification, and second-order optimality conditions for tame convex optimization problems, extending classical results to a broader class.

Findings

Optimal solutions are generically unique and on a single manifold.

Feasible regions are partly smooth around solutions, aiding algorithmic identification.

Second-order conditions ensure solution stability under perturbations.

Abstract

We consider linear optimization over a fixed compact convex feasible region that is semi-algebraic (or, more generally, "tame"). Generically, we prove that the optimal solution is unique and lies on a unique manifold, around which the feasible region is "partly smooth", ensuring finite identification of the manifold by many optimization algorithms. Furthermore, second-order optimality conditions hold, guaranteeing smooth behavior of the optimal solution under small perturbations to the objective.