# Mathematical programs with complementarity constraints: convergence properties of a smoothing method

G. Bouza Allende, Georg J. Still

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15 Citations (Scopus)

## Abstract

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 $\cal{O}(\sqrt{\tau})$.
Original language Undefined 10.1287/moor.1060.0245 467-483 17 Mathematics of operations research 32 LNCS4549/2 https://doi.org/10.1287/moor.1060.0245 Published - May 2007

• EWI-11089
• MSC-65K05
• IR-64357
• MSC-90C33
• METIS-241917
• MSC-90C30