Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares
Year: 2015
Type: article
Abstract: In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, 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 $H_\infty$ 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.
Authors: Reza Kamyar, Matthew M. Peet
Institution: Arizona State University
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Awards: Solving Large Sum-of-Squares Optimization Problems in Control by Exploiting the Parallel Structure of Polya's Algorithm, CAREER: A New Computational Framework for Control of Complex Systems, Solving Large Sum-of-Squares Optimization Problems in Control by Exploiting the Parallel Structure of Polya's Algorithm
OpenAlex ID: W2126658944