Hadamard Directional Differentiability of the Optimal Value Function of a Quadratic Programming Problem

Abstract

In this paper, we consider the stability analysis of a convex quadratic programming (QP) problem and its restricted Wolfe dual when all parameters in the problem are perturbed. Based on the continuity of the feasible set mapping, we establish the upper semi-continuity of the optimal solution mappings of the convex QP problem and the restricted Wolfe dual problem. Furthermore, by characterizing the optimal value function as a min–max optimization problem over two compact convex sets, we demonstrate the Lipschitz continuity and the Hadamard directional differentiability of the optimal value function.