Hessian-Free Methods for Checking the Second-Order Sufficient Conditions in Equality-Constrained Optimization and Equilibrium Problems

TL;DR

This paper introduces three efficient Hessian-free methods for verifying the Second-Order Sufficient Condition in constrained optimization, enabling detection of negative curvature directions without explicit Hessian computation.

Contribution

The paper proposes three novel Hessian-free tests for SOSC verification that rely solely on gradient evaluations, improving efficiency and providing negative curvature directions when SOSC fails.

Findings

New Hessian-free tests are computationally efficient.

Classical tests require explicit Hessian knowledge.

Comparative analysis shows improved efficiency of new methods.

Abstract

Verifying the Second-Order Sufficient Condition (SOSC), thus ensuring a stationary point locally minimizes a given objective function (subject to certain constraints), is an essential component of non-convex computational optimization and equilibrium programming. This article proposes three new "Hessian-free" tests of the SOSC that can be implemented efficiently with gradient evaluations alone and reveal feasible directions of negative curvature when the SOSC fails. The Bordered Hessian Test and a Matrix Inertia test, two classical tests of the SOSC, require explicit knowledge of the Hessian of the Lagrangian and do not reveal feasible directions of negative curvature should the SOSC fail. Computational comparisons of the new methods with classical tests demonstrate the relative efficiency of these new algorithms and the need for careful study of false negatives resulting from accumulation of round-off errors.