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

作者：Jeyakumar V^*; Lee G M; Lee J H

来源：Journal of Convex Analysis, 2015, 22(4): 999-1023.

• Strong duality
• rho-SOS-convex polynomials
• SOS-convex polynomials
• non-convex quadratic optimization
• extended trust-region problems

In this paper, we introduce the notion of rho-SOS-convexity, extending the numerically checkable concept of SOS-convexity of a real polynomial. The class of rho-SOS-convex polynomials includes the important class of (not necessarily convex) quadratic functions. We provide various characterizations of rho-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 rho > 0) and weak SOS-convex (where rho < 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.

• 出版日期2015