On the Generation of Positivstellensatz Witnesses in Degenerate Cases

David Monniaux; Pierre Corbineau

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arXiv · 1105.4421

On the Generation of Positivstellensatz Witnesses in Degenerate Cases

David Monniaux·Pierre Corbineau

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Abstract

One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq. The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty. We implemented our method and illustrate its use with examples, including extractions of proofs to Coq.

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BibTeX RIS

David Monniaux, Pierre Corbineau. 2011-05-23. On the Generation of Positivstellensatz Witnesses in Degenerate Cases. https://arxiv.org/abs/1105.4421

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