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The tracial moment problem and trace-positive polynomialsBurgdorf, Sabine ...Themain topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f , what is the smallest trace f (A) can attain for a tuple of ... matrices A? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives effectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satisfies a certain condition called flatness, then the relaxation is exact. In this case it is shown how to extract global trace-optimizers with a procedure based on two ingredients. The first is the solution to the truncated tracial moment problem, and the other crucial component is the numerical implementation of the Artin-Wedderburn theorem for matrix -algebras due to Murota, Kanno, Kojima, Kojima, and Maehara. Trace-optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other sidetwo topics with many applications, the most prominent being to linear systems engineering and quantum physics. The optimization problems discussed here facilitate new possibilities for applications, e.g. in operator algebras and statistical physics.Source: Mathematical programming. - ISSN 0025-5610 (Vol. 137, iss. 1/2, Feb. 2013, str. 557-578)Type of material - article, component partPublish date - 2013Language - englishCOBISS.SI-ID - 2048170515
Author
Burgdorf, Sabine |
Cafuta, Kristijan, 1977- |
Klep, Igor, matematik |
Povh, Janez, 1973-
Topics
sum of squares |
noncommutative polynomial |
semidefinite programming |
tracial moment problem |
flat extension |
free positivity |
real algebraic geometry |
vsota kvadratov |
nekomutativni polinom |
semidefinitno programiranje |
problem momenta sledi |
ploska razširitev |
prosta pozitivnost |
realna algebraična geometrija
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Database name | Field | Year |
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Burgdorf, Sabine | |
Cafuta, Kristijan, 1977- | 28255 |
Klep, Igor, matematik | 22353 |
Povh, Janez, 1973- | 22649 |
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