Abstract
Motivated by robust (non-convex) quadratic optimization over convex quadratic constraints, in this paper, we examine minimax difference of convex (dc) optimization over convex polynomial inequalities. By way of generalizing the celebrated Farkas’ lemma to inequality systems involving the maximum of dc functions and convex polynomials, we show that there is no duality gap between a minimax DC polynomial program and its associated conjugate dual problem. We then obtain strong duality under a constraint qualification. Consequently, we present characterizations of robust solutions of uncertain general non-convex quadratic optimization problems with convex quadratic constraints, including uncertain trust-region problems.
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The authors are grateful to the referees for their comments and suggestions which have contributed to the final preparation of the paper. Research of the first author was partially supported by a grant from the Australian Research Council, whereas the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2005378).
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Jeyakumar, V., Lee, G.M. & Linh, N.T.H. Generalized Farkas’ lemma and gap-free duality for minimax DC optimization with polynomials and robust quadratic optimization. J Glob Optim 64, 679–702 (2016). https://doi.org/10.1007/s10898-015-0277-4
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DOI: https://doi.org/10.1007/s10898-015-0277-4