Mathematical Programming

, Volume 137, Issue 1–2, pp 557–578

The tracial moment problem and trace-optimization of polynomials

  • Sabine Burgdorf
  • Kristijan Cafuta
  • Igor Klep
  • Janez Povh
Full Length Paper Series A

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.

Keywords

Sum of squares Noncommutative polynomial Semidefinite programming Tracial moment problem Flat extension Free positivity Real algebraic geometry 

Mathematics Subject Classification (2000)

Primary 90C22 13J30 Secondary 47A57 08B20 

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Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  • Sabine Burgdorf
    • 1
  • Kristijan Cafuta
    • 2
  • Igor Klep
    • 3
    • 4
  • Janez Povh
    • 5
  1. 1.Institut de MathématiquesUniversité de NeuchâtelNeuchâtelSuisse
  2. 2.Fakulteta za elektrotehniko, Laboratorij za uporabno matematikoUniverza v LjubljaniLjubljanaSlovenia
  3. 3.Fakulteta za naravoslovje in matematikoUniverza v MariboruMariborSlovenia
  4. 4.Fakulteta za matematiko in fizikoUniverza v LjubljaniLjubljanaSlovenia
  5. 5.Fakulteta za informacijske študije v Novem mestuNovo mestoSlovenia

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