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Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones

Akiko Yoshise (yoshise***at***sk.tsukuba.ac.jp)

Abstract: We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in Faybusovich (1997) depends on the optimization theory of convex log-barrier functions, our approach is based on the paper of Monteiro and Pang (1998), where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices. As an application of the results, we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects. Consequently, we show the existence of a path having the following properties: (a) The path is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions. (b) Any accumulation point of the path is a solution of the homogeneous model. (c) If the original problem is solvable, then every accumulation point of the path gives us a finite solution. (d) If the original problem is strongly infeasible, then, under the assumption of Lipschitz continuity, any accumulation point of the path gives us a finite certificate proving infeasibility.

Keywords: Symmetric cone, Complementarity problem, Homogeneous algorithm, Existence of trajectory, Interior point method, Detecting infeasibility

Category 1: Linear, Cone and Semidefinite Programming (Other )

Category 2: Complementarity and Variational Inequalities

Citation: SIAM Journal on Optimization 17 (2006) 1129 - 1153

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Entry Submitted: 08/09/2004
Entry Accepted: 08/09/2004
Entry Last Modified: 01/16/2007

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