Sums of Separable and Quadratic Polynomials

TL;DR

This paper investigates the structure and nonnegativity conditions of sums of separable and quadratic polynomials, providing decomposition results, complexity insights, and applications in optimization and statistics.

Contribution

It characterizes when nonnegative SPQ polynomials can be decomposed into sums of nonnegative parts and explores their applications in optimization and other fields.

Findings

Positive decomposition for univariate plus quadratic and convex SPQ polynomials.

NP-hardness of testing nonnegativity for degree ≥ 4 SPQ polynomials.

Applications in polynomial optimization, regression, and Newton's method.

Abstract

We study separable plus quadratic (SPQ) polynomials, i.e., 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, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial, and (ii) a sum of squares. We 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. We use our decomposition result for convex SPQ polynomials to show that convex SPQ polynomial optimization problems can be solved by "small" semidefinite programs. For question (ii), we provide a complete characterization of the answer based on the degree and the number of variables of the SPQ polynomial. We also prove that testing nonnegativity of SPQ polynomials is NP-hard when the degree is at least four. We 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 which incorporates separable higher-order derivative information.