Sums of Separable and Quadratic Polynomials

The authors study separable plus quadratic (SPQ) polynomials, that is, polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sums of squares, the authors study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial and (ii) a sum of squares. The authors establish that the answer to question (i) is positive for univariate plus quadratic polynomials and for convex SPQ polynomials but negative already for bivariate quartic SPQ polynomials. The authors use their decomposition result for convex SPQ polynomials to show that convex SPQ polynomial optimization problems can be solved by “small” semidefinite programs. For question (ii), the authors provide a complete characterization of the answer based on the degree and the number of variables of the SPQ polynomial. They also prove that testing nonnegativity of SPQ polynomials is NP-hard when the degree is at least four. The authors end by presenting applications of SPQ polynomials to upper bounding sparsity of solutions to linear programs, polynomial regression problems in statistics, and a generalization of Newton’s method that incorporates separable higher order derivative information.