A New Smoothing Method for Mathematical Programs with Complementarity Constraints Based on Logarithm-Exponential Function

Abstract

We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC). Different from the existing smoothing methods available in the literature, we construct an approximate smooth problem of MPCC by partly smoothing the complementarity constraints. With this new method, it is proved that the Mangasarian-Fromovitz constraint qualification holds for the approximate smooth problem. Convergence of the approximate solution sequence, generated by solving a series of smooth perturbed subproblems, is investigated. Under the weaker constraint qualification MPCC-Cone-Continuity Property, it is proved that any accumulation point of the approximate solution sequence is a M-stationary point of the original MPCC. Preliminary numerical results indicate that the developed algorithm based on the partly smoothing method is efficient, particularly in comparison with the other similar ones.

A Smoothing Newton Method with a Mixed Line Search for Monotone Weighted Complementarity Problems

“…is is the so-called smoothing Newton method. is class of methods has been successfully applied to solving lots of optimization and related problems, including linear complementarity problems [19][20][21][22][23][24], linear programs [25], nonlinear complementarity problems [26][27][28][29][30], variational inequalities [27,31], semidefinite complementarity problems [32][33][34][35][36][37], system of inequalities [19,34], symmetric cone complementarity problems [35,36], mathematical programs with complementarity constraints [38,39], and absolute value equations [40]. In order to achieve the global convergence, most of the known smoothing Newton methods require the assumption that the solution set of the concerned problem is nonempty and compact.…”