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

Vaithilingam Jeyakumar () and Guoyin Li ()
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Vaithilingam Jeyakumar: University of New South Wales
Guoyin Li: University of New South Wales

Journal of Optimization Theory and Applications, 2017, vol. 172, issue 1, No 9, 156-178

Abstract: 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.

Keywords: Convex cone programs; Convex polynomial optimization; Conic programming; Semidefinite programming; 90C22; 90C25; 90C46 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10957-016-1023-x

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