Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone Programs

In this paper, under a suitable regularity condition, we establish that a broad class of conic convex polynomial optimization problems, called conic sum-of-squares convex polynomial programs, exhibits exact conic programming relaxation, which can be solved by various numerical methods such as interior point methods. By considering a general convex cone-program, we give unified results that apply to many classes of important cone-programs, such as the second-order cone programs, semidefinite programs and polyhedral cone programs. When the cones involved in the programs are polyhedral cones we present a regularity-free exact semidefinite programming relaxation. We do Vaithilingam Jeyakumar University of New South Wales, Sydney 2052, Australia E-mail: [email protected] Corresponding Author: Guoyin Li University of New South Wales, Sydney 2052, Australia E-mail: [email protected] 2 Vaithilingam Jeyakumar, Guoyin Li this by establishing a sum-of-squares polynomial representation of positivity of a real sums-of-squares convex polynomial over a conic sums-of-squares-convex system. In many cases, the sum-of-squares representation can be numerically checked via solving a conic programming problem. Consequently, we also show that a convex set, described by a conic sums-of-squares-convex polynomial, is (lifted) conic linear representable in the sense that it can be expressed as (a projection of) the set of solutions to some conic linear systems.