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Trace Optimization of Polynomials in Non-commuting Variables

  • Sabine Burgdorf
  • Igor Klep
  • Janez Povh
Chapter
  • 461 Downloads
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In Chap.  3 trace-positivity together with the question how to detect it was explored in details. Due to hardness of the decision problem “Is a given nc polynomial f trace-positive?” we proposed a relaxation of the problem, i.e., we are asking if f is cyclically equivalent to SOHS. The tracial Gram matrix method based on the tracial Newton polytope was proposed (see Sects. 3.3 and 3.4) to efficiently detect such polynomials.

Keywords

Trace Optimization Semidefinite Programming Feasibility Problem Flat Solutions Archimedean Quadratic Module Tracer Version 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [CKP11]
    Cafuta, K., Klep, I., Povh, J.: NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials. Optim. Methods. Softw. 26 (3), 363–380 (2011). Available from http://ncsostools.fis.unm.si/ MathSciNetCrossRefzbMATHGoogle Scholar
  2. [Con76]
    Connes, A.: Classification of injective factors. Cases II1, II\(_{\infty },\) III\(_{\lambda },\) \(\lambda \not =1\). Ann. Math. (2) 104 (1), 73–115 (1976)Google Scholar
  3. [Ji13]
    Ji, Z.: Binary Constraint System Games and Locally Commutative Reductions. arXiv preprint. arXiv:1310.3794 (2013)Google Scholar
  4. [KP16]
    Klep, I., Povh, J.: Constrained trace-optimization of polynomials in freely noncommuting variables. J. Glob Optim. 64, 325–348 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [KS08]
    Klep, I., Schweighofer, M.: Connes’ embedding conjecture and sums of hermitian squares. Adv. Math. 217 (4), 1816–1837 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Las01]
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11 (3), 796–817 (2000/01)Google Scholar
  7. [Las09]
    Lasserre, J.B.: Moments, Positive Polynomials and Their Application. Imperial College Press, London (2009)CrossRefzbMATHGoogle Scholar
  8. [LP15]
    Laurent, M., Piovesan, T.: Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone. SIAM J. Optim. 25 (4), 2461–2493 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Sabine Burgdorf
    • 1
  • Igor Klep
    • 2
  • Janez Povh
    • 3
  1. 1.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  2. 2.Department of MathematicsThe University of AucklandAucklandNew Zealand
  3. 3.Faculty of Information Studies in Novo MestoNovo MestoSlovenia

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