A D-Induced Duality and Its Applications

Jan Brinkhuis (brinkhuisfew.eur.nl)
Shuzhong Zhang (zhangse.cuhk.edu.hk)

Abstract: This paper attempts to extend the notion of duality for convex cones, by basing it on a pre-described conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the {\em nonnegativity}\/ of the inner-product is replaced by a pre-specified conic ordering, defined by a convex cone $D$, and the inner-product itself is replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed the {\em $D$-induced duality}\/ in the paper. Basic properties of the extended duality, including the extended bi-polar theorem, are proven. Examples are given to show the applications of the new results.

Keywords: convex cones, duality, bi-polar theorem, conic optimization

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: Technical Report SEEM2002-01,Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, 2002.

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Entry Submitted: 09/20/2002
Entry Accepted: 09/21/2002
Entry Last Modified: 09/20/2002

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