Convergence of a local regularization approach for mathematical programmes with complementarity or vanishing constraints

Abstract

Mathematical programmes with equilibrium or vanishing constraints (MPECs or MPVCs) are both known to be difficult optimization problems which typically violate all standard constraint qualifications. A number of methods try to exploit the particular structure of MPECs and MPVCs in order to overcome these difficulties. In a recent paper by Steffensen and Ulbrich (S. Steffensen and M. Ulbrich, A new regularization scheme for mathematical programs with equilibrium constraints, SIAM J. Optim. 2010.), this was done for MPECs by a local regularization idea that may be viewed as a modification of the popular global regularization technique by Scholtes (S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM J. Optim. 11 (2001), pp. 918–936.). The aim of this paper is twofold. First, we improve the convergence theory from (S. Steffensen and M. Ulbrich, A new regularization scheme for mathematical programs with equilibrium constraints, SIAM J. Optim. 2010.) in the MPEC setting, and second we translate this local regularization idea to MPVCs and obtain a new solution method for this class of optimization problems for which several convergence results are given.

Keywords:

• mathematical programmes with equilibrium constraints
• mathematical programmes with vanishing constraints
• regularization method
• global convergence

Acknowledgements

The second author would like to thank Gabriel Haeser from the University of Campinas for pointing his attention to the CPLD condition as a new constraint qualification for nonlinear programmes. The authors also wish to thank the referees for their helpful comments leading to a significant improvement of a previous version of this paper.

This research was partially supported by a grant from the international doctorate programme ‘Identification, Optimization, and Control with Applications in Modern Technologies’ within the Elite-Network of Bavaria.