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Constrained trace-optimization of polynomials in freely noncommuting variables

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Abstract

The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry. In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation scheme is governed by flatness, i.e., a rank-preserving property for associated dual SDPs. If flatness is observed, then optimizers can be extracted using the Gelfand–Naimark–Segal construction and the Artin–Wedderburn theory verifying exactness of the relaxation. To enforce flatness we employ a noncommutative version of the randomization technique championed by Nie. The implementation of these procedures in our computer algebra system NCSOStoolsis presented and several examples are given to illustrate our results.

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Authors and Affiliations

  1. Department of Mathematics, The University of Auckland, Auckland, New Zealand

    Igor Klep

  2. Faculty of information studies in Novo mesto, Novo mesto, Slovenia

    Janez Povh

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  1. Igor Klep
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  2. Janez Povh
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Correspondence to Janez Povh.

Additional information

Igor Klep: Supported by the Marsden Fund Council of the Royal Society of New Zealand. Partially supported by the Slovenian Research Agency Grants P1-0222, L1-4292 and L1-6722. Part of this research was done while the author was on leave from the University of Maribor.

Janez Povh: Supported by the Slovenian Research Agency via program P1-0383 and project L7-4119. Supported by the Creative Core FISNM-3330-13-500033 ‘Simulations’ project funded by the European Union.

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Klep, I., Povh, J. Constrained trace-optimization of polynomials in freely noncommuting variables. J Glob Optim 64, 325–348 (2016). https://doi.org/10.1007/s10898-015-0308-1

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