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

TL;DR

This paper extends Yuan's theorem of the alternative to tensor settings, enabling polynomial optimization problems with sign constraints to be solved efficiently via convex conic programming and sum-of-squares relaxation.

Contribution

The paper introduces a tensor-based extension of Yuan's theorem, linking nonconvex polynomial optimization with convex conic programming and sum-of-squares methods.

Findings

Optimal values of certain polynomial problems are computable via convex conic programming.

Exact sum-of-squares relaxation applies to these polynomial problems.

Solutions can be recovered from convex conic problem solutions.

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 non-positive 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.