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Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems

Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland 21218-2682
Department of Mathematics, National University of Singapore, Singapore
Department of Decision Sciences and Singapore-MIT Alliance, National University of Singapore, Singapore
pang(mts.jhu.edu
matsundf(nus.edu.sg
jsun(nus.edu.sg

Based on an inverse function theorem for a system of semismooth equations, this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular/stable. We show further that for a parametric complementarity problem of this kind, if a solution corresponding to a base parameter is strongly stable, then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution. Similar results are also derived for a complementarity problem defined on the Lorentz cone. The analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the Lorentz cone.

Key Words: Complementarity problem; variational inequality; semidefinite cone; Lorentz cone
History: Received: April 14, 2002; revision received: October 12, 2002;

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