Optimization with SOS Polynomials

Abstract

This chapter describes the use of SOS polynomials in typical optimization problems over polynomials. First, the case of unconstrained optimization is considered. It is shown how one can establish whether a polynomial is either positive or non-positive by introducing a generalized SOS index which measures how a polynomial is the ratio of two SOS polynomials and which can be found by solving an SDP. Then, the problem of determining the minimum of a rational function is considered, showing how one can obtain a lower bound through the generalized SOS index. This lower bound converges to the sought minimum as the order of the generalized SOS index increases, moreover a necessary and sufficient condition is provided for establishing its tightness. Second, the case of constrained optimization is considered. Stengle Positivstellensatz and some of its re-elaborations are reviewed, which provide strategies for establishing positivity of a polynomial over semialgebraic sets based on SOS polynomials. Then, the problem of determining the minimum of a rational function over semialgebraic sets is considered. For this problem it is shown that a direct use of the Positivstellensatz allows one to obtain a lower bound through SDPs, moreover a necessary and sufficient condition is provided for establishing its tightness. Moreover, the problem of solving systems of polynomial equations and inequalities is addressed, showing how a combination of the generalized SOS index and the Positivstellensatz can be used to compute the sought solutions through an SDP. Also, the problem of establishing whether a matrix polynomial is positive definite over semialgebraic sets is addressed by introducing a matrix version of the Positivstellensatz. Third, the case of optimization over special sets is considered. In particular, the chapter addresses the problems of establishing positivity of a polynomial over an ellipsoid and over the simplex. It is shown that these problems are equivalent to establishing positivity of a homogeneous polynomial constructed without introducing unknown multipliers. Lastly, the problem of searching for a positive semidefinite SMR matrix of a SOS polynomial with the largest rank is considered, showing that it can be exactly solved with a finite sequence of LMI feasibility tests.