The tracial moment problem and trace-optimization of polynomials

Burgdorf, Sabine; Cafuta, Kristijan; Klep, Igor; Povh, Janez

The tracial moment problem and trace-optimization of polynomials

Dateien

287 Burgdorf.pdfGröße: 795.32 KBDownloads: 369

Datum

2011

Autor:innen

Burgdorf, Sabine

Cafuta, Kristijan

Klep, Igor

Povh, Janez

Publikationstyp

Working Paper/Technical Report

Zusammenfassung

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(A) can attain for a tuple of matrices A? A relaxation using semide nite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, it gives e ectively computable bounds on the optima. To test for exactness, the solution of the dual SDP is investigated. If it satis es a certain condition called atness, 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.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

sum of squares, noncommutative polynomial, semide nite programming, tracial moment problem, at extension, free positivity, real algebraic geometry

Zitieren

ISO 690

BURGDORF, Sabine, Kristijan CAFUTA, Igor KLEP, Janez POVH, 2011. The tracial moment problem and trace-optimization of polynomials

BibTex

@techreport{Burgdorf2011traci-15284,
  year={2011},
  series={Konstanzer Schriften in Mathematik},
  title={The tracial moment problem and trace-optimization of polynomials},
  number={287},
  author={Burgdorf, Sabine and Cafuta, Kristijan and Klep, Igor and Povh, Janez}
}

RDF

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