  In this paper, we explore the merits of various algorithms for polynomial
optimization 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, our
main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and
Handelman's theorem. We first formulate polynomial optimization problems as
verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's
algorithm, Bernstein's algorithm and Handelman's algorithm reduce the
intractable problem of feasibility of semi-algebraic sets to linear and/or
semi-definite programming. We apply these algorithms to different problems in
robust stability analysis and stability of nonlinear dynamical systems. As one
contribution of this paper, we apply Polya's algorithm to the problem of
H_infinity control of systems with parametric uncertainty. Numerical examples
are provided to compare the accuracy of these algorithms with other polynomial
optimization algorithms in the literature.
