Sums of Separable and Quadratic Polynomials
This work addresses foundational questions in polynomial optimization with applications in linear programming and statistics, though it is incremental in extending known results to SPQ structures.
The paper tackled the problem of characterizing nonnegative sums of separable and quadratic (SPQ) polynomials, establishing that convex SPQ polynomials can be decomposed into nonnegative separable and quadratic parts, enabling solution via small semidefinite programs, and proving NP-hardness for testing nonnegativity when degree is at least four.
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.