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Constrained polynomial optimization problems with noncommuting variablesCafuta, Kristijan, 1977- ; Klep, Igor, matematik ; Povh, Janez, 1973-In this paper we study constrained eigenvalue optimization of noncommutative (nc) polynomials, focusing on the polydisc and the ball. Our three main results are as follows: (1) an nc polynomial is ... nonnegative if and only if it admits a weighted sum of hermitian squares decomposition; (2) (eigenvalue) optima for nc polynomials can be computed using a single semidefinite program (SDP) - this sharply contrasts with the commutative case where sequences of SDPs are needed; (3) the dual solution to this "single" SDP can be exploited to extract eigenvalue optimizers with an algorithm based on two ingredients: solution to a truncated nc moment problem via flat extensions, and Gelfand-Naimark-Segal construction. The implementation of these procedures in our computer algebra system NCSOStools is presented, and several examples pertaining to matrix inequalities are given to illustrate our results.Source: SIAM journal on optimization. - ISSN 1052-6234 (Vol. 22, no. 2, 2012, str. 363-383)Type of material - article, component partPublish date - 2012Language - englishCOBISS.SI-ID - 16276569
Author
Cafuta, Kristijan, 1977- |
Klep, Igor, matematik |
Povh, Janez, 1973-
Topics
nekomutativni polinom |
optimizacija |
vsota kvadratov |
semidefinitno programiranje |
problem momentov |
Hankelova matrika |
realna algebraična geometrija |
prosta pozitivnost |
noncommutative polynomial |
optimization |
sum of squares |
semidefinite programming |
moment problem |
Hankel matrix |
flat extension |
MATLAB toolbox |
real algebraic geometry |
free positivity
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Cafuta, Kristijan, 1977- | 28255 |
Klep, Igor, matematik | 22353 |
Povh, Janez, 1973- | 22649 |
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