About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 130682, 14 pages
http://dx.doi.org/10.1155/2012/130682
Research Article

Lipschitz Continuity of the Solution Mapping of Symmetric Cone Complementarity Problems

1Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China
2Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan
3Mathematics Division, National Center for Theoretical Sciences (Taipei Office), Taipei 10617, Taiwan

Received 24 February 2012; Accepted 25 August 2012

Academic Editor: Malisa R. Zizovic

Copyright © 2012 Xin-He Miao and Jein-Shan Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the Lipschitz continuity of the solution mapping of symmetric cone (linear or nonlinear) complementarity problems (SCLCP or SCCP, resp.) over Euclidean Jordan algebras. We show that if the transformation has uniform Cartesian P-property, then the solution mapping of the SCCP is Lipschitz continuous. Moreover, we establish that the monotonicity of mapping and the Lipschitz continuity of solutions of the SCLCP imply ultra P-property, which is a concept recently developed for linear transformations on Euclidean Jordan algebra. For a Lyapunov transformation, we prove that the strong monotonicity property, the ultra P-property, the Cartesian P-property, and the Lipschitz continuity of the solutions are all equivalent to each other.