A D-induced duality and its applications

Jan Brinkhuis and Shuzhong Zhang

No EI 2003-42, Econometric Institute Research Papers from Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute

Abstract: This paper attempts to extend the notion of duality for convex cones, by basing it on a predescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the 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 D-induced duality in the paper. We further introduce the notion of D-induced polar sets within the same framework, which can be viewed as a generalization of the D-induced polar sets within the same framework, which can be viewed as a generalization of the D-induced dual cones and are convenient to use for some practical applications. Properties of the extended duality, including the extended bi-polar theorem, are proven. Furthermore, attention is paid to the computation and approximation of the D-induced dual objects. We discuss, as examples, applications of the newly introduced D-induced duality concepts in robust conic optimization and the duality theory for multi-objective conic optimization.

Keywords: bi-polar theorem; conic optimization; convex cones; duality (search for similar items in EconPapers)
Date: 2003-08-07
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