Gram Matrix Representation

Abstract

There are several ways of characterizing nonnegative polynomials that may be interesting for a mathematician. However, not all of them are appropriate for computational purposes, by “computational” understanding primarily optimization methods. Nonnegative polynomials have a basic property extremely useful in optimization: They form a convex set. So, an optimization problem whose variables are the coefficients of a nonnegative polynomial has a unique solution (or, in the degenerate case, multiple solutions belonging to a convex set), if the objective and the other constraints besides positivity are also convex. Convexity is not enough for obtaining efficiently a reliable solution. Efficiency and reliability are specific only to some classes of convex optimization, such as linear programming (LP), second-order cone problems (SOCP), and semidefinite programming (SDP). SDP includes LP and SOCP and is probably the most important advance in optimization in the last decade of the previous century. See some basic information on SDP in Appendix A. In this chapter, we present a parameterization of nonnegative polynomials that is intimately related to SDP. Each polynomial can be associated with a set of matrices, called Gram matrices (Choi et al., Proc Symp Pure Math 58:103–126, 1995, [1]); if the polynomial is nonnegative, then there is at least a positive semidefinite Gram matrix associated with it. Solving optimization problems with nonnegative polynomials may thus be reduced, in many cases, to SDP. We give several examples of such problems and of programs that solve them. Spectral factorization is important in this context, and we present several techniques for its computation. Besides the standard, or trace, parameterization, we discuss several other possibilities that may have computational advantages.