Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems

Authors:
Jong-Shi Pang;Defeng Sun;Jie Sun
Affiliations:
Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland;Department of Mathematics, National University of Singapore, Singapore;Department of Decision and Singapore and Singapore-MIT Alliance, National University of Singapore, Singapore
Venue:
Mathematics of Operations Research
Year:
2003

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Abstract

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.