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The tracial moment problem and trace-optimization of polynomials

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Abstract

The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace \({f(\underline {A})}\) can attain for a tuple of matrices \({\underline {A}}\)? 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 side—two 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.

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Authors and Affiliations

  1. Institut de Mathématiques, Université de Neuchâtel, 11 rue Emile-Argand, 2000, Neuchâtel, Suisse

    Sabine Burgdorf

  2. Fakulteta za elektrotehniko, Laboratorij za uporabno matematiko, Univerza v Ljubljani, Tržaška cesta 25, 1000, Ljubljana, Slovenia

    Kristijan Cafuta

  3. Fakulteta za naravoslovje in matematiko, Univerza v Mariboru, Koroška cesta 160, 2000, Maribor, Slovenia

    Igor Klep

  4. Fakulteta za matematiko in fiziko, Univerza v Ljubljani, Jadranska ulica 19, 1111, Ljubljana, Slovenia

    Igor Klep

  5. Fakulteta za informacijske študije v Novem mestu, Novi trg 5, 8000, Novo mesto, Slovenia

    Janez Povh

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  1. Sabine Burgdorf
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  2. Kristijan Cafuta
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  3. Igor Klep
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  4. Janez Povh
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Correspondence to Janez Povh.

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Burgdorf, S., Cafuta, K., Klep, I. et al. The tracial moment problem and trace-optimization of polynomials. Math. Program. 137, 557–578 (2013). https://doi.org/10.1007/s10107-011-0505-8

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