A Sequential Smooth Penalization Approach to Mathematical Programs with Complementarity Constraints

In this paper, a mathematical program with complementarity constraints (MPCC) is reformulated as a nonsmooth constrained mathematical program via the Fischer–Burmeister function. Smooth penalty functions are used to treat this nonsmooth constrained program. Under linear independence constraint qualification, and upper level strict complementarity condition, together with some other mild conditions, we prove that the limit point of stationary points satisfying second-order necessary conditions of unconstrained penalized problems is a strongly stationary point, hence a B-stationary point of the original MPCC. Furthermore, this limit point also satisfies a second-order necessary condition of the original MPCC. Numerical results are presented to test the performance of this method.

Keywords:

• B-stationary point
• Complementarity constraints
• Linear independence constraint qualification
• Mathematical program
• Optimality condition
• Penalty function

AMS Classification:

• 90C30
• 90C33
• 90C26

ACKNOWLEDGMENTS

The authors are grateful to Dr. Danny Ralph for some useful comments and suggestions on the paper and for providing us with the QPECgen package. We also thank Mr. Xinmin Hu for sending us the manuscript of his paper co-authored with Dr. Danny Ralph while the manuscript of this paper was being revised. This work was supported by the Research Grants Council of Hong Kong (Poly B-Q359).