Numerical Verification of Second-Order Sufficiency Conditions for Nonlinear Programming

The existence of second-order sufficient conditions (SOSCs) for nonlinear programming problems is an interesting theoretical result. It is not, however, immediately clear how one might use them in practice to determine if a candidate point produced by an algorithm operating on an actual problem is indeed optimal. We present a simple method for computationally checking the SOSCs (for the case when the Karush-Kuhn-Tucker (KKT) multipliers for all active inequality constraints are positive) in a finite number of steps that at once sheds light on their meaning and provides a computational tool for use in checking the results produced by algorithms on actual problems. Furthermore, this method serves as a nice bridge from a class on the theory of nonlinear programming to one on algorithms, as it connects the SOSCs to the conjugate structure of a problem, a concept important in the construction of many solution methods.