A SMOOTHING METHOD FOR MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

摘要

This paper studies a smoothing method for mathematical programs with complementarity constraints (MPCC) based on the integral of the sigmoid function, in which the original MPCC is reformulated as a standard smooth approximation minimization problem with a smooth parameter and the approximate solution of the MPCC is obtained by solving a series of the smooth subproblems where the smooth parameter approaches zero. It is proven that the accumulation point of the sequence of KKT solutions to the smooth subproblems is a C-stationary point of the MPCC under the MPCC-MFCQ, without the upper level strict complementarity and the asymptotically weakly nondegenerate condition. Furthermore, the accumulation point can be proven to be an S-stationary point under the weak second-order necessary condition. Moreover, the characterizations of the linear independence constraints qualification, the KKT condition and the second-order sufficient condition for the smooth approximation problem are established under several assumptions on the original MPCC, which ensures the existence of KKT solutions to the smooth subproblems. At last, the numerical experiments are implemented to test the performance of the smoothing method by solving some typical problems in MacMPEC database. The reported numerical results show that the smoothing method is promising by comparing with those by the other typical methods in the existent references.