Convergence Properties of a Smoothing Approach for Mathematical Programs with Second-Order Cone Complementarity Constraints

Abstract

This paper discusses the convergence properties of a smoothing approach for solving the mathematical programs with second-order cone complementarity constraints (SOCMPCCs). We first introduce B-stationary, C-stationary, M(orduckhovich)-stationary, S-stationary point, SOCMPCC-linear independence constraint qualification (denoted by SOCMPCC-LICQ), second-order cone upper level strict complementarity (denoted by SOC-ULSC) condition at a feasible point of a SOCMPCC problem. With the help of the projection operator over a second-order cone, we construct a smooth optimization problem to approximate the SOCMPCC. We demonstrate that any accumulation point of the sequence of stationary points to the sequence of smoothing problems, when smoothing parameters decrease to zero, is a C-stationary point to the SOCMPCC under SOCMPCC-LICQ at the accumulation point. We also prove that the accumulation point is an M-stationary point if, in addition, the sequence of stationary points satisfy weak second order necessary conditions for the sequence of smoothing problems, and moreover it is a B-stationary point if, in addition, the SOC-ULSC condition holds at the accumulation point.