Error bounds for symmetric cone complementarity problems

Xin-He Miao; Jein-Shan Chen

doi:10.3934/naco.2013.3.627

Article Contents

2013, Volume 3, Issue 4: 627-641. Doi: 10.3934/naco.2013.3.627

This issue Previous Article Some properties of a class of $(F,E)$-$G$ generalized convex functions Next Article Characterizations of the $E$-Benson proper efficiency in vector optimization problems

Error bounds for symmetric cone complementarity problems

Xin-He Miao^1, and
Jein-Shan Chen^2,

1.
Department of Mathematics, School of Science, Tianjin University, Tianjin 300072
2.
Department of Mathematics, National Taiwan Normal University, Taipei 11677

Received: May 2013

Revised: August 2013

Published: October 2013

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Abstract

In this paper, we investigate the issue of error bounds for symmetric cone complementarity problems (SCCPs). In particular, we show that the distance between an arbitrary point in Euclidean Jordan algebra and the solution set of the symmetric cone complementarity problem can be bounded above by some merit functions such as Fischer-Burmeister merit function, the natural residual function and the implicit Lagrangian function. The so-called $R_0$-type conditions, which are new and weaker than existing ones in the literature, are assumed to guarantee that such merit functions can provide local and global error bounds for SCCPs. Moreover, when SCCPs reduce to linear cases, we demonstrate such merit functions cannot serve as global error bounds under general monotone condition, which implicitly indicates that the proposed $R_0$-type conditions cannot be replaced by $P$-type conditions which include monotone condition as special cases.

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Mathematics Subject Classification: Primary: 65K10; Secondary: 90C33.

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