Gram Matrix Representation

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