SOSOPT: A Toolbox for Polynomial Optimization

SOSOPT is a Matlab toolbox for formulating and solving Sum-of-Squares (SOS) polynomial optimizations. This document briefly describes the use and functionality of this toolbox. Section 1 introduces the problem formulations for SOS tests, SOS feasibility problems, SOS optimizations, and generalized SOS problems. Section 2 reviews the SOSOPT toolbox for solving these optimizations. This section includes information on toolbox installation, formulating constraints, solving SOS optimizations, and setting optimization options. Finally, Section 3 briefly reviews the connections between SOS optimizations and semidefinite programs (SDPs). It is the connection to SDPs that enables SOS optimizations to be solved in an efficient manner. 1 Sum of Squares Optimizations This section describes several optimizations that can be formulated with sum-of-squares (SOS) polynomials [14, 11, 15]. A multivariable polynomial is a SOS if it can be expressed as a sum of squares of other polynomials. In other words, a polynomial p is SOS if there exists polynomials {fi} m i=1 such that p = ∑m i=1 f 2 i . An SOS polynomial is globally nonnegative because each squared term is nonnegative. This fact enables sufficient conditions for many analysis problems to be posed as optimizations with polynomial SOS constraints. This includes many nonlinear analysis problems such as computing regions of attraction, reachability sets, input-output gains, and robustness with respect to uncertainty for nonlinear polynomial systems [14, 25, 7, 9, 8, 15, 17, 12, 6, 4, 22, 10, 13, 21, 23, 26, 30, 29, 27, 24, 28, 1]. The remainder of this section defines SOS tests, SOS feasibility problems, SOS optimizations, and generalized SOS optimizations. Given a polynomial p(x), a sum-of-squares test is an analysis problem of the form: