On the computation of relaxed pessimistic solutions to MPECs

Abstract

In this paper, we propose a new numerical method to compute approximate and the so-called relaxed pessimistic solutions to mathematical programs with equilibrium constraints (MPECs), where the solution map arising in the equilibrium constraints is not single-valued. This method combines two types of existing codes, a code for derivative-free optimization under box constraints, BFO or BOBYQA, and a method for solving special parametric MPECs from the interactive system UFO. We report on numerical performance in several small-dimensional test problems.

Keywords:

• MPEC
• equilibrium constraints
• pessimistic solution
• value function
• relaxed and approximate solutions

Acknowledgements

We are grateful to M.J.D. Powell and Ph.L. Toint for their kind help in providing the codes of BOBYQA and BFO. We also thank M. Kočvara for numerous fruitful suggestions and discussions and to Alexandra Schwartz for a careful reading of preliminary versions of our manuscript. The authors would also like to thank the anonymous referees for their careful review of the manuscript and for many constructive comments and suggestions. This work was supported by the Grant Agency of the Czech Republic, project No. 201/09/1957, by the Ministry of Education of the Czech Republic, grant 1M0572, and by the institutional research plan No. AV0Z10300504.