A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign structure

Shenglong Hu (), Guoyin Li () and Liqun Qi ()
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Shenglong Hu: Tianjin University
Guoyin Li: University of New South Wales
Liqun Qi: The Hong Kong Polytechnic University

Journal of Optimization Theory and Applications, 2016, vol. 168, issue 2, No 4, 446-474

Abstract: Abstract Yuan’s theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan’s theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of a class of nonconvex polynomial optimization problems with suitable sign structure (or more explicitly, with essentially nonpositive coefficients) can be computed by a related convex conic programming problem, and the optimal solution of these nonconvex polynomial optimization problems can be recovered from the corresponding solution of the convex conic programming problem. Moreover, we obtain that this class of nonconvex polynomial optimization problems enjoy exact sum-of-squares relaxation, and so, can be solved via a single semidefinite programming problem.

Keywords: Alternative theorem; Symmetric tensors; Nonconvex polynomial optimization; Sum-of-squares relaxation; Semidefinite programming; 90C26; 90C22; 15A69 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10957-014-0652-1

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