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On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets

On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets

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KRIEL, Tom-Lukas, Markus SCHWEIGHOFER, 2017. On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets

@techreport{Kriel2017-04-24T13:50:50Zexact-38753, title={On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets}, year={2017}, author={Kriel, Tom-Lukas and Schweighofer, Markus} }

eng Schweighofer, Markus Kriel, Tom-Lukas 2017-04-24T13:50:50Z Consider a finite system of non-strict real polynomial inequalities and suppose its solution set S⊆R<sup>n</sup> is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of S. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on S (which is very natural because of the convexity of S) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on S (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation.<br />In their seminal work of 2009, Helton and Nie showed under the same conditions that S is the projection of a spectrahedron, i.e., it has a semidefinite representation. The semidefinite representation used by Helton and Nie arises from glueing together Lasserre relaxations of many small pieces obtained in a non-constructive way. By refining and varying their approach, we show that we can simply take a Lasserre relaxation of the original system itself. Such a result was provided by Helton and Nie with much more machinery only under very technical conditions and after changing the description of S. Schweighofer, Markus 2017-05-05T13:33:43Z On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets 2017-05-05T13:33:43Z Kriel, Tom-Lukas

Dateiabrufe seit 05.05.2017 (Informationen über die Zugriffsstatistik)

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