Geodesic
Generalized SOS-Convexity and Strong Duality with SDP Dual Programs in Polynomial Optimization
Journal of convex analysis, Tome 22 (2015) no. 4, pp. 999-1023
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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.
Classification : 26A51, 90C25, 47N10
Mots-clés : Strong duality, rho-SOS-convex polynomials, SOS-convex polynomials, non-convex quadratic optimization, extended trust-region problems 
@article{JCA_2015_22_4_JCA_2015_22_4_a5,
     author = {V. Jeyakumar and G. M. Lee and J. H. Lee},
     title = {Generalized {SOS-Convexity} and {Strong} {Duality} with {SDP} {Dual} {Programs} in {Polynomial} {Optimization}},
     journal = {Journal of convex analysis},
     pages = {999--1023},
     year = {2015},
     volume = {22},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_4_JCA_2015_22_4_a5/}
}
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V. Jeyakumar; G. M. Lee; J. H. Lee. Generalized SOS-Convexity and Strong Duality with SDP Dual Programs in Polynomial Optimization. Journal of convex analysis, Tome 22 (2015) no. 4, pp. 999-1023. http://geodesic.mathdoc.fr/item/JCA_2015_22_4_JCA_2015_22_4_a5/