Error Bounds for Piecewise Convex Quadratic Programs and Applications

Abstract

In this paper, we establish a local error estimate for feasible solutions of a piecewise convex quadratic program and a global error estimate for feasible solutions of a convex piecewise quadratic program. These error estimates provide a unified approach for deriving many old and new error estimates for linear programs, linear complementarity problems, convex quadratic programs, and affine variational inequality problems. The approach reveals the fact that each error estimate is a consequence of some reformulation of the original problem as a piecewise convex quadratic program or a convex piecewise quadratic program. In a sense, even Robinson's result on the upper Lipschitz continuity of a polyhedral mapping can be considered as a special case of error estimates for approximate solutions of a piecewise convex quadratic program. As an application, we derive new (global) error estimates for iterates of the proximal point algorithm for solving a convex piecewise quadratic program.