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 \({\mathsf{D}}\) , and the inner-product itself is replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed the \({\mathsf{D}}\)-induced duality in the paper. We further introduce the notion of \({\mathsf{D}}\)-induced polar sets within the same framework, which can be viewed as a generalization of the \({\mathsf{D}}\)-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 \({\mathsf{D}}\) -induced dual objects. We discuss, as examples, applications of the newly introduced \({\mathsf{D}}\)-induced duality concepts in robust conic optimization and the duality theory for multi-objective conic optimization.
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References
Ben-Tal A., Nemirovskii A. (1998). Robust convex optimization. Math. Oper. Res. 23: 769–805
Brinkhuis J., Tikhomirov V. (2005). Optimization: insights and applications. Princeton University Press, Princeton
El Ghaoui L., Lebret H. (1997). Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18: 1035–1064
Grötschel M., Lovász L., Schrijver A. (1993). Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg
Luo, Z.Q., Sturm, J.F., Zhang, S.: Duality results for conic convex programming. Technical Report 9719/A, Econometric Institute, Erasmus University Rotterdam, The Netherlands (1997)
Luo Z.Q., Sturm J.F., Zhang S. (2004). Multivariate nonnegative quadratic mappings. SIAM J. Optim. 14: 1140–1162
Nemirovskii A., Roos C., Terlaky T. (1999). On maximization of quadratic form over intersection of ellipsoids with common center. Math. Progr. 86: 463–473
Nesterov Yu. (1998). Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Software 9: 141–160
Nesterov, Yu.: Global quadratic optimization on the sets with simple structure. Working Paper, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (1999)
Nesterov, Yu., Nemirovsky, A.: Interior point polynomial methods in convex programming. Stud. Appl. Math. 13 (1994)
Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton
Sturm J.F., Zhang S. (2003). On cones of nonnegative quadratic functions. Math. Oper. Res. 28: 246–267
Ye Y. (1999). Approximating quadratic programming with bound and quadratic constraints. Math. Progr. 84: 219–226
Zhang S. (2004). A new self-dual embedding method for convex programming. J. Global Optim. 29: 479–496
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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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Brinkhuis, J., Zhang, S. A \({\mathsf{D}}\)-induced duality and its applications. Math. Program. 114, 149–182 (2008). https://doi.org/10.1007/s10107-007-0097-5
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DOI: https://doi.org/10.1007/s10107-007-0097-5