Partial Augmented Lagrangian method and Mathematical Programs with Complementarity Constraints^{1}

X. X. Huang, X. Q. Yang and K. L. Teo

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

Abstract. In this paper, we apply a partial augmented Lagrangian method to mathematical programs with complementarity constraints (MPCC). Specifically, only the complementarity constraints are incorporated into the objective function of the augmented Lagrangian problem while the other constraints of the original (MPCC) are retained as constraints in the augmented Lagrangian problem. We show that the limit point of a sequence of points that satisfy second-order necessary conditions of the partial augmented Lagrangian problems is a strongly stationary point (hence a B-stationary point) of the original (MPCC) if the limit point is feasible to (MPCC), the linear independence constraint qualification for (MPCC) and the upper level strict complementarity condition hold at the limit point. Furthermore, this limit point also satisfies a second-order necessary optimality condition of (MPCC). Numerical results are reported to show the effectiveness of this partial augmented Lagrangian method for (MPCC).

Keywords: Partial augmented Lagrangian method, mathematical programs with complementarity constraints, constraint qualification, optimality conditions, B-stationarity.

1This research is supported by the Research Grants Council of Hong Kong.