A Smoothing Method for Zero–One Constrained Extremum Problems

Abstract

In this paper, the zero–one constrained extremum problem is reformulated as an equivalent smooth mathematical program with complementarity constraints (MPCC), and then as a smooth ordinary nonlinear programming problem with the help of the Fischer–Burmeister function. The augmented Lagrangian method is adopted to solve the resulting problem, during which the non-smoothness may be introduced as a consequence of the possible inequality constraints. This paper incorporates the aggregate constraint method to construct a uniform smooth approximation to the original constraint set, with approximation controlled by only one parameter. Convergence results are established, showing that under reasonable conditions the limit point of the sequence of stationary points generated by the algorithm is a strongly stationary point of the original problem and satisfies the second order necessary conditions of the original problem. Unlike other penalty type methods for MPCC, the proposed algorithm can guarantee that the limit point of the sequence is feasible to the original problem.