Abstract
In this paper, under a suitable regularity condition, we establish a broad class of conic convex polynomial optimization problems, called conic sum-of-squares convex polynomial programs, exhibiting 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 this by establishing a sum-of-squares polynomial representation of positivity of a real sum-of-squares convex polynomial over a conic sum-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 sum-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.
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The authors would like to express their sincere thanks to the associate editor and the referee for their constructive comments and valuable suggestions which have contributed to the revision of this paper.
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Communicated by Levent Tunçel.
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Jeyakumar, V., Li, G. Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone Programs. J Optim Theory Appl 172, 156–178 (2017). https://doi.org/10.1007/s10957-016-1023-x
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DOI: https://doi.org/10.1007/s10957-016-1023-x