Publikation: Constrained polynominal optimization problems with noncommuting variables
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2011
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Zusammenfassung
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 semide nite program (SDP) { this sharply contrasts 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 at extensions; Gelfand-Naimark-Segal (GNS) 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.
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510 Mathematik
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noncommutative polynomial, optimization, sum of squares, semide nite programming, moment problem, Hankel matrix, at extension, Matlab toolbox, real algebraic geometry, free positivity
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CAFUTA, Kristijan, Igor KLEP, Janez POVH, 2011. Constrained polynominal optimization problems with noncommuting variablesBibTex
@techreport{Cafuta2011Const-15283,
year={2011},
series={Konstanzer Schriften in Mathematik},
title={Constrained polynominal optimization problems with noncommuting variables},
number={285},
author={Cafuta, Kristijan and Klep, Igor and Povh, Janez}
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