Dual Semidefinite Programs Without Duality Gaps for a Class of Convex Minimax Programs

V. Jeyakumar () and J. Vicente-Pérez ()
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V. Jeyakumar: University of New South Wales
J. Vicente-Pérez: University of New South Wales

Journal of Optimization Theory and Applications, 2014, vol. 162, issue 3, No 4, 735-753

Abstract: Abstract In this paper, we introduce a new dual program, which is representable as a semidefinite linear programming problem, for a primal convex minimax programming problem, and we show that there is no duality gap between the primal and the dual whenever the functions involved are sum-of-squares convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust sum-of-squares convex programming problems under data uncertainty and to minimax fractional programming problems with sum-of-squares convex polynomials. We obtain these results by first establishing sum-of-squares polynomial representations of non-negativity of a convex max function over a system of sum-of-squares convex constraints. The new class of sum-of-squares convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The sum-of-squares convexity of polynomials can numerically be checked by solving semidefinite programming problems whereas numerically verifying convexity of polynomials is generally very hard.

Keywords: Sum-of-squares convex polynomials; Minimax programming; Semidefinite programming; Duality; Zero duality gap (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (8)

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DOI: 10.1007/s10957-013-0496-0

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