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In this paper, we explore the merits of various algorithms for polynomialoptimization problems, focusing on alternatives to sum of squares programming.While we refer to advantages and disadvantages of Quantifier Elimination,Reformulation Linear Techniques, Blossoming and Groebner basis methods, ourmain focus is on algorithms defined by Polya's theorem, Bernstein's theorem andHandelman's theorem. We first formulate polynomial optimization problems asverifying the feasibility of semi-algebraic sets. Then, we discuss how Polya'salgorithm, Bernstein's algorithm and Handelman's algorithm reduce theintractable problem of feasibility of semi-algebraic sets to linear and/orsemi-definite programming. We apply these algorithms to different problems inrobust stability analysis and stability of nonlinear dynamical systems. As onecontribution of this paper, we apply Polya's algorithm to the problem ofH_infinity control of systems with parametric uncertainty. Numerical examplesare provided to compare the accuracy of these algorithms with other polynomialoptimization algorithms in the literature.

Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares