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Kristijan Cafuta (kristijan.cafuta***at***fe.uni-lj.si)
Igor Klep (igor.klep***at***fmf.uni-lj.si)
Janez Povh (janez.povh***at***fis.unm.si)

Abstract: 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 flat 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.

Keywords: noncommutative polynomial, optimization, sum of squares, semide

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Optimization Software and Modeling Systems (Problem Solving Environments )

Citation: SIAM Journal on Optimization, 2012, vol. 22, no. 2, pp. 363-383.

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Entry Submitted: 03/21/2011
Entry Accepted: 03/21/2011
Entry Last Modified: 04/25/2012

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