2011/06/05 by William R. Morrow, Morrow, W. Ross
Computer Science · Mathematics · #65 #90 #91 #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.1106.0898
openalex publication_date 2011/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.