SONC optimization and exact nonnegativity certificates via second-order cone programming. (March 2023)

• Record Type: Journal Article
• Authors: Magron, Victor; Wang, Jie
• Is Part Of: Journal of symbolic computation. Volume 115(2023)
• Page Start: 346
• Page End: 370
• Publication Date: 2023-03
• Subjects: Sum of nonnegative circuit polynomials -- Second-order cone programming -- Polynomial optimization -- Sum of binomial squares -- Rounding-projection algorithm -- Exact nonnegativity certificate; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.08.002
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables.