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∞ 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.2010 Mathematics Subject Classification. Primary: 93D05, 93D09; Secondary: 90C22.

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In this paper, we propose a distributed computing approach to solving large-scale robust stability problems on the simplex. Our approach is to formulate the robust stability problem as an optimization problem with polynomial variables and polynomial inequality constraints. We use Polya's theorem to convert the polynomial optimization problem to a set of highly structured Linear Matrix Inequalities (LMIs). We then use a slight modification of a common interior-point primaldual algorithm to solve the structured LMI constraints. This yields a set of extremely large yet structured computations. We then map the structure of the computations to a decentralized computing environment consisting of independent processing nodes with a structured adjacency matrix. The result is an algorithm which can solve the robust stability problem with the same per-core complexity as the deterministic stability problem with a conservatism which is only a function of the number of processors available. Numerical tests on cluster computers and supercomputers demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors and analyze systems with 100+ dimensional state-space. The proposed algorithms can be extended to perform stability analysis of nonlinear systems and robust controller synthesis. Index Terms-Robust stability, Polynomial optimization, Large-scale systems, Decentralized computingReza Kamyar received the B.S. and M.S in aerospace engineering from Sharif University of Technology, Tehran, Iran in 2008, and 2010. He is currently a Ph.D student in the department of mechanical engineering of Arizona State University, Tempe, Arizona. He is a research assistant with Cybernetic Systems and Controls Laboratory (CSCL) in the School for Engineering of Matter, Transport and Energy (SEMTE) at Arizona State University. His research focuses on the development of decentralized algorithms applied to the problems of stability and control of large-scale complex systems.

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