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Monotonicity of L\"{o}wner Operators and Its Applications to Symmetric Cone Complementarity Problems

Lingchen Kong(konglchen***at***126.com)
Levent Tuncel(ltuncel***at***math.uwaterloo.ca)
Naihua Xiu(nhxiu***at***bjtu.edu.cn)

Abstract: This paper focuses on monotone L\"{o}wner operators in Euclidean Jordan algebras and their applications to the symmetric cone complementarity problem (SCCP). We prove necessary and sufficient conditions for locally Lipschitz L\"{o}wner operators to be monotone, strictly monotone and strongly monotone. We also study the relationship between monotonicity and operator-monotonicity of L\"{o}wner operators. As a by-product of our results, we establish a new class of C-functions for SCCP, which is an extension of the Mangasarian class of NCP-functions for the nonlinear complementarity problem, and present some characterizations of the C-functions for SCCP under certain assumptions.

Keywords: L\"{o}wner operator, Euclidean Jordan algebra, Monotonicity, Symmetric cone complementarity problem, C-function

Category 1: Complementarity and Variational Inequalities

Category 2: Convex and Nonsmooth Optimization (Generalized Convexity/Monoticity )

Category 3: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

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Entry Submitted: 04/16/2007
Entry Accepted: 04/17/2007
Entry Last Modified: 04/16/2007

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