Approximation Hierarchies for Copositive Tensor Cone

Iqbal, Muhammad Faisal; Ahmed, Faizan; Aqeel, Muhammad; Ahmad, Sahrim

doi:10.48550/arxiv.1806.01063

Cited by 1 publication

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“…Optimization over the cone of copositive polynomials is hard [54]; however, this cone has been well studied. In particular, there exists a plenty of tractable approximations to it [see, e.g., 12,34,49,51,60,89], as well as several certificates of copositivity [for instance , 21, 29]. The main benefit of our copositive certificates of non-negativity is their ability to translate results known exclusively for copositive polynomials to more general semialgebraic sets (see Section 1.2 for a more detailed explanation of our contributions).…”

Section: Introductionmentioning

confidence: 99%

The copositive way to obtain certificates of non-negativity over general semialgebraic sets

Kuryatnikova¹,

Vera²,

Zuluaga³

2019

Preprint

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A certificate of non-negativity is a way to write a given function so that its non-negativity becomes evident. Certificates of non-negativity are fundamental tools in optimization, and they underlie powerful algorithmic techniques for various types of optimization problems. We propose certificates of non-negativity of polynomials based on copositive polynomials. The certificates we obtain are valid for generic semialgebraic sets and have a fixed small degree, while commonly used sums-of-squares (SOS) certificates are guaranteed to be valid only for compact semialgebraic sets and could have large degree. Optimization over the cone of copositive polynomials is not tractable, but this cone has been well studied. The main benefit of our copositive certificates of non-negativity is their ability to translate results known exclusively for copositive polynomials to more general semialgebraic sets. In particular, we show how to use copositive polynomials to construct structured (e.g., sparse) certificates of non-negativity, even for unstructured semialgebraic sets. Last but not least, copositive certificates can be used to obtain not only hierarchies of tractable lower bounds, but also hierarchies of tractable upper bounds for polynomial optimization problems.

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