2019•arXiv (Cornell University)Open access

Second-order cone representations of SONC cones

Jie Wang

Abstract

The second-order cone is a class of simple convex cones and the optimization problem over second-order cones can be solved more efficiently than semidefinite programming. Given that second-order cones have a strong expressive ability, it is interesting to investigate which convex cones admit a representation using second-order cones. In this paper, we prove that all SONC cones surprisingly admit a second-order cone representation, which is dramatically different from the case of positive semidefinite cones and SOS cones as Fawzi very recently proved that the $3\times3$ positive semidefinite cone does not admit any second-order cone representation. Based on this, we give a new formulization of SONC optimization via second-order cone programming.

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Key concepts: Cone (formal languages), Second-order cone programming, Conic optimization, Semidefinite programming, Representation (politics), Regular polygon, Mathematics, Order (exchange)