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

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  • Vaithilingam Jeyakumar

    (University of New South Wales)

  • Guoyin Li

    (University of New South Wales)

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.

Suggested Citation

  • Vaithilingam Jeyakumar & Guoyin Li, 2017. "Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone Programs," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 156-178, January.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:1:d:10.1007_s10957-016-1023-x
    DOI: 10.1007/s10957-016-1023-x
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    References listed on IDEAS

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    1. Jiawang Nie, 2011. "Polynomial Matrix Inequality and Semidefinite Representation," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 398-415, August.
    2. V. Jeyakumar & J. B. Lasserre & G. Li, 2014. "On Polynomial Optimization Over Non-compact Semi-algebraic Sets," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 707-718, December.
    3. V. Jeyakumar & D.T. LUC, 2008. "Nonsmooth Vector Functions and Continuous Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-73717-1, December.
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    Cited by:

    1. T. D. Chuong & V. Jeyakumar & G. Li, 2019. "A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs," Journal of Global Optimization, Springer, vol. 75(4), pages 885-919, December.

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