Journal
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
Volume 162, Issue 3, Pages 735-753Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10957-013-0496-0
Keywords
Sum-of-squares convex polynomials; Minimax programming; Semidefinite programming; Duality; Zero duality gap
Funding
- Australian Research Council
- MICINN of Spain [MTM2011-29064-C03-02]
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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.
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