  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.
