A {mathsf{D}}-induced duality and its applications |
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Authors: | Jan Brinkhuis Shuzhong Zhang |
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Affiliation: | (1) Econometric Institute, Erasmus University, Rotterdam, The Netherlands;(2) Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong |
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Abstract: | This paper attempts to extend the notion of duality for convex cones, by basing it on a prescribed 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 , and the inner-product itself is replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed the -induced duality in the paper. We further introduce the notion of -induced polar sets within the same framework, which can be viewed as a generalization of the -induced dual cones and is 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 -induced dual objects. We discuss, as examples, applications of the newly introduced -induced duality concepts in robust conic optimization and the duality theory for multi-objective conic optimization. Research supported in part by the Foundation ‘Vereniging Trustfonds Erasmus Universiteit Rotterdam’ in The Netherlands, and in part by Hong Kong RGC Earmarked Grants CUHK4174/03E and CUHK418406. |
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Keywords: | Convex cones Duality Duality theorem Conic optimization |
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