Generic identifiability and second-order sufficiency in tame convex optimization
Provides theoretical guarantees for optimization algorithms on a broad class of convex problems, relevant to optimization theory and applications.
The paper proves that for generic linear optimization over tame convex sets, the optimal solution is unique and lies on a unique manifold, ensuring finite identification by algorithms and smooth behavior under perturbations.
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.