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Journal of Convex Analysis 22 (2015), No. 4, 999--1023
Copyright Heldermann Verlag 2015

Generalized SOS-Convexity and Strong Duality with SDP Dual Programs in Polynomial Optimization

Vaithilingam Jeyakumar
Dept. of Applied Mathematics, University of New South Wales, Sydney 2052, Australia

Gue Myung Lee
Dept. of Applied Mathematics, Pukyong National University, Busan 608-737, Korea

Jae Hyoung Lee
Dept. of Applied Mathematics, Pukyong National University, Busan 608-737, Korea

We introduce the notion of ρ-SOS-convexity, extending the numerically checkable concept of SOS-convexity of a real polynomial. The class of ρ-SOS-convex polynomials includes the important class of (not necessarily convex) quadratic functions. We provide various characterizations of ρ-SOS-convexity in terms of SOS-convexity. Consequently, we establish strong duality results for classes of nonconvex polynomial optimization problems involving strong SOS-convex (where ρ > 0) and weak SOS-convex (where ρ < 0) polynomials. These classes of problems include some polynomial optimization problems, involving SOS-convex polynomials, minimax quadratic optimization problems with quadratic constraints, fractional programming problems and robust optimization problems. Our results also provide necessary and sufficient conditions for strong duality of some classes of minimax quadratic optimization problems and extended trust-region problems.

Keywords: Strong duality, rho-SOS-convex polynomials, SOS-convex polynomials, non-convex quadratic optimization, extended trust-region problems.

MSC: 26A51, 90C25, 47N10

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