2021/05/11 by Amir Ali Ahmadi, Ahmadi, Amir Ali, Cemil Dibek +3 · 1 citation
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Numerical Methods and Algorithms #Optimization and Control (math.OC) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2105.04766
openalex publication_date 2021/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.