Solution point differentiability without strict complementarity in nonlinear programming

Abstract

In an earlier study [4], we have established that for the local optimal solution point of a nonlinear programming problem to be stable with respect to small perturbations only the linear independence and a strong second order condition are needed. The strict complementarity condition is not. The optimal solution of the perturbed problem is a unique continuous function of the perturbation parameter. In this study, we extend the results in [4] by investigating, without strict complementarity, differentiability properties of the optimal solution point with respect to the perturbation parameter. It is possible to show that, even though the optimal solution point may not be differentiable with respect to the perturbation parameter, its directional derivative with respect to any directional perturbation always exists and is the unique solution of a system of equations and inequalities. A possible method for computing the directional derivative is also suggested.