On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints

Abstract

This paper focuses on a class of mathematical programs with symmetric cone complementarity constraints (SCMPCC). The explicit expression of C-stationary condition and SCMPCC-linear independence constraint qualification (denoted by SCMPCC-LICQ) for SCMPCC are first presented. We analyze a parametric smoothing approach for solving this program in which SCMPCC is replaced by a smoothing problem \begin{document}$P_{\varepsilon}$\end{document} depending on a (small) parameter \begin{document}$\varepsilon$\end{document} . We are interested in the convergence behavior of the feasible set, stationary points, solution mapping and optimal value function of problem \begin{document}$P_{\varepsilon}$\end{document} when \begin{document}$\varepsilon \to 0$\end{document} under SCMPCC-LICQ. In particular, it is shown that the convergence rate of Hausdorff distance between feasible sets \begin{document}$\mathcal{F}_{\varepsilon}$\end{document} and \begin{document}$\mathcal{F}$\end{document} is of order \begin{document}$\mbox{O}(|\varepsilon|)$\end{document} and the solution mapping and optimal value of \begin{document}$P_{\varepsilon}$\end{document} are outer semicontinuous and locally Lipschitz continuous at \begin{document}$\varepsilon=0$\end{document} respectively. Moreover, any accumulation point of stationary points of \begin{document}$P_{\varepsilon}$\end{document} is a C-stationary point of SCMPCC under SCMPCC-LICQ.