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Mathematical Programs with Complementarity Constraints: Convergence Properties of a Smoothing Method

Department of Mathematics, University of Havana, Vedado, Plaza CP:10440, Havana, Cuba
Department of Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
gema{at}matcom.uh.cu
g.still{at}math.utwente.nl

In this paper, optimization problems P with complementarity constraints are considered. Characterizations for local minimizers \bar{x} of P of Orders 1 and 2 are presented. We analyze a parametric smoothing approach for solving these programs in which P is replaced by a perturbed problem P_{\tau} depending on a (small) parameter \tau. We are interested in the convergence behavior of the feasible set \cal{F}_{\tau} and the convergence of the solutions \bar{x}_{\tau} of P_{\tau} for \tau\to 0. In particular, it is shown that, under generic assumptions, the solutions \bar{x}_{\tau} are unique and converge to a solution \bar{x} of P with a rate \cal{O}(\sqrt{\tau}). Moreover, the convergence for the Hausdorff distance d(\cal{F}_{\tau}, \cal{F}) between the feasible sets of P_{\tau} and P is of order \cal{O}(\sqrt{\tau}).

Key Words: mathematical programs with complementarity constraints; smoothing method; rate of convergence; genericity
History: Received: December 7, 2004; revision received: May 17, 2006;