Sums of Hermitian squares decomposition of non-commutative polynomials in non-symmetric variables using NCSOStools

  • Kristijan CafutaEmail author
Original Paper


Numerous applied problems contain matrices as variables, and the formulas therefore involve polynomials in matrices. To handle such polynomials it is necessary study non-commutative polynomials. In this paper we will present an algorithm and its implementation in the free Matlab package NCSOStools using semidefinite programming to check whether a given non-commutative polynomial in non-symmetric variables can be written as a sum of Hermitian squares.


Noncommutative polynomial Sum of Hermitian squares Semidefinite programming Matlab toolbox NCSOStools 

Mathematics Subject Classification

13J30 90C22 08B20 11E25 90C90 



The author thanks both anonymous referees for helpful suggestions.


  1. Anjos M, Lasserre J (2012) Handbook of semidefinite, conic and polynomial optimization: theory, algorithms, software and applications, vol 166. International series in operational research and management science. Springer, New YorkCrossRefGoogle Scholar
  2. Bachoc C, Gijswijt DC, Schrijver A, Vallentin F (2012) Invariant semidefinite programs. In: Handbook on semidefinite, conic and polynomial optimization, international series in operations research & management science, vol 166. Springer, New York, pp 219–269Google Scholar
  3. Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. Studies in applied mathematics. SIAM, PhiladelphiaCrossRefGoogle Scholar
  4. Burgdorf S, Cafuta K, Klep I, Povh J (2013a) Algorithmic aspects of sums of hermitian squares of noncommutative polynomials. Comput Optim Appl 55(1):137–153CrossRefGoogle Scholar
  5. Burgdorf S, Cafuta K, Klep I, Povh J (2013b) The tracial moment problem and trace-optimization of polynomials. Math Program 137(1):557–578CrossRefGoogle Scholar
  6. Burgdorf S, Klep I, Povh J (2016) Optimization of polynomials in non-commuting variables. Springer briefs in mathematics. Springer, BerlinGoogle Scholar
  7. Cafuta K (2013) On matrix algebras associated to sum-of-squares semidefinite programs. Linear Multilinear Algebra 61(11):1496–1509CrossRefGoogle Scholar
  8. Cafuta K, Klep I, Povh J (2010) A note on the nonexistence of sum of squares certificates for the Bessis–Moussa–Villani conjecture. J Math Phys 51(8):083,521, 10CrossRefGoogle Scholar
  9. Cafuta K, Klep I, Povh J (2011) NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials. Optim Methods Softw 26(3):363–380.
  10. Cafuta K, Klep I, Povh J (2012) Constrained polynomial optimization problems with noncommuting variables. SIAM J Optim 22(2):363–383CrossRefGoogle Scholar
  11. Cafuta K, Klep I, Povh J (2015) Rational sums of hermitian squares of free noncommutative polynomials. Ars Math Contemp 9(2):243–259CrossRefGoogle Scholar
  12. Choi M, Lam T, Reznick B (1995) Sums of squares of real polynomials. In: Proceedings of symposia in pure mathematics, \(K\)-theory and algebraic geometry: connections with quadratic forms and division algebras, vol 58. AMS, Providence, pp 103–126Google Scholar
  13. Cimprič J (2010) A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming. J Math Anal Appl 369(2):443–452CrossRefGoogle Scholar
  14. de Oliveira M, Helton J, McCullough S, Putinar M (2008) Engineering systems and free semi-algebraic geometry. In: Emerging applications of algebraic geometry, IMA voume of mathematics and its application, vol 149. Springer, New York, pp 17–61Google Scholar
  15. Gatermann K, Parrilo P (2004) Symmetry groups, semidefinite programs, and sums of squares. J Pure Appl Algebra 192(1–3):95–128CrossRefGoogle Scholar
  16. Goemans MX (1997) Semidefinite programming in combinatorial optimization. Math Program 79(1–3, Ser. B):143–161Google Scholar
  17. Halická M, de Klerk E, Roos C (2002) On the convergence of the central path in semidefinite optimization. SIAM J Optim 12(4):1090–1099CrossRefGoogle Scholar
  18. Helton J (2002) “Positive” noncommutative polynomials are sums of squares. Ann of Math 156(2):675–694CrossRefGoogle Scholar
  19. Helton J, Nie J (2012) A semidefinite approach for truncated K-moment problems. Found Comput Math 12(6):851–881CrossRefGoogle Scholar
  20. Horn R, Johnson C (1985) Matrix analysis. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  21. Klep I, Povh J (2010) Semidefinite programming and sums of hermitian squares of noncommutative polynomials. J Pure Appl Algebra 214:740–749CrossRefGoogle Scholar
  22. Klep I, Povh J (2016) Constrained trace-optimization of polynomials in freely noncommuting variables. J Global Optim 64(2):325–348CrossRefGoogle Scholar
  23. Klep I, Schweighofer M (2008a) Connes’ embedding conjecture and sums of Hermitian squares. Adv Math 217(4):1816–1837CrossRefGoogle Scholar
  24. Klep I, Schweighofer M (2008b) Sums of Hermitian squares and the BMV conjecture. J Stat Phys 133(4):739–760CrossRefGoogle Scholar
  25. Lasserre J (2000/2001) Global optimization with polynomials and the problem of moments. SIAM J Optim 11(3):796–817Google Scholar
  26. Lasserre J (2009) Moments, positive polynomials and their applications, vol 1. Imperial college press optimization series. Imperial College Press, LondonGoogle Scholar
  27. Laurent M (2009) Sums of squares, moment matrices and optimization over polynomials. In: Emerging applications of algebraic geometry, IMA volumes in mathematics and its application, vol 149. Springer, New York, pp 157–270Google Scholar
  28. Marshall M (2008) Positive polynomials and sums of squares, mathematical surveys and monographs, vol 146. American Mathematical Society, ProvidenceCrossRefGoogle Scholar
  29. McCullough S (2001) Factorization of operator-valued polynomials in several non-commuting variables. Linear Algebra Appl 326(1–3):193–203CrossRefGoogle Scholar
  30. McCullough S, Putinar M (2005) Noncommutative sums of squares. Pac J Math 218(1):167–171CrossRefGoogle Scholar
  31. Nie J (2009) Sum of squares method for sensor network localization. Comput Optim Appl 43(2):151–179CrossRefGoogle Scholar
  32. Nie J (2014) Optimality conditions and finite convergence of Lasserre’s hierarchy. Math Program 146(1–2):97–121CrossRefGoogle Scholar
  33. Parrilo P (2000) Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD. thesis, California Institute of TechnologyGoogle Scholar
  34. Parrilo P (2003) Semidefinite programming relaxations for semialgebraic problems. Math Program 96(2, Ser. B):293–320CrossRefGoogle Scholar
  35. Pironio S, Navascués M, Acín A (2010) Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM J Optim 20(5):2157–2180CrossRefGoogle Scholar
  36. Powers V, Wörmann T (1998) An algorithm for sums of squares of real polynomials. J Pure Appl Algebra 127(1):99–104CrossRefGoogle Scholar
  37. Recht B, Fazel M, Parrilo P (2010) Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev 52(3):471–501CrossRefGoogle Scholar
  38. Reznick B (1978) Extremal PSD forms with few terms. Duke Math J 45(2):363–374CrossRefGoogle Scholar
  39. Shor NZ (1991) Dual quadratic estimates in polynomial and boolean programming. Ann Oper Res 25(1–4):163–168Google Scholar
  40. Sturm J (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11/12(1–4):625–653.
  41. Toh K, Todd M, Tütüncü R (2012) On the implementation and usage of SDPT3—a Matlab software package for semidefinite-quadratic-linear programming, version 4.0. In: Handbook on semidefinite, conic and polynomial optimization, international series in operations research & management science, vol 166. Springer, New York, pp 715–754.
  42. Wolkowicz H, Saigal R, Vandenberghe L (2000) Handbook of semidefinite programming. Kluwer, DordrechtCrossRefGoogle Scholar
  43. Yamashita M, Fujisawa K, Kojima M (2003) Implementation and evaluation of SDPA 6.0 (semidefinite programming algorithm 6.0). Optim Methods Softw 18(4):491–505.

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratorij za uporabno matematiko in statistiko, Fakulteta za elektrotehnikoUniverza v LjubljaniLjubljanaSlovenia

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