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A D-induced duality and its applications

Author

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  • Brinkhuis, J.
  • Zhang, S.

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.

Suggested Citation

  • Brinkhuis, J. & Zhang, S., 2003. "A D-induced duality and its applications," Econometric Institute Research Papers EI 2003-42, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:1058
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    References listed on IDEAS

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    1. Luo, Z-Q & Sturm, J.F. & Zhang, S., 2003. "Multivariate Nonnegative Quadratic Mappings," Discussion Paper 2003-7, Tilburg University, Center for Economic Research.
    2. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
    3. Luo, Z-Q & Sturm, J.F. & Zhang, S., 2003. "Multivariate Nonnegative Quadratic Mappings," Other publications TiSEM 77619783-0422-424b-8dc7-2, Tilburg University, School of Economics and Management.
    4. Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1997. "Duality Results for Conic Convex Programming," Econometric Institute Research Papers EI 9719/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
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