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. 

　　Publication: 

　　Journal of Symbolic Computation, 2023, Volume 115, pp: 346-370. 

　　Author: 

　　Victor Magron 

　　CNRS LAAS, 7 avenue du colonel Roche 31400 Toulouse, France 

　　Institute of Mathematics, 31400 Toulouse, France 

　　Wang Jie 

　　Academy of Mathematics and Systems Science, CAS, China 

　　Email: wangjie212@amss.ac.cn