Journal of Global Optimization

, Volume 64, Issue 2, pp 325–348 | Cite as

Constrained trace-optimization of polynomials in freely noncommuting variables



The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry. In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation scheme is governed by flatness, i.e., a rank-preserving property for associated dual SDPs. If flatness is observed, then optimizers can be extracted using the Gelfand–Naimark–Segal construction and the Artin–Wedderburn theory verifying exactness of the relaxation. To enforce flatness we employ a noncommutative version of the randomization technique championed by Nie. The implementation of these procedures in our computer algebra system NCSOStoolsis presented and several examples are given to illustrate our results.


Noncommutative polynomial Optimization Sum of squares Semidefinite programming Moment problem Hankel matrix Flat extension Matlab toolbox  Real algebraic geometry Free positivity 

Mathematics Subject Classification

Primary 90C22 14P10 Secondary 13J30 47A57 


  1. 1.
    Anjos, M.F., Lasserre, J.B.: Handbook of Semidefinite, Conic and Polynomial Optimization: Theory, Algorithms, Software and Applications, volume 166 of International Series in Operational Research and Management Science. Springer, Berlin (2012)CrossRefGoogle Scholar
  2. 2.
    Barvinok, A.: A Course in Convexity, Volume 54 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2002)Google Scholar
  3. 3.
    Burgdorf, S., Cafuta, K., Klep, I., Povh, J.: The tracial moment problem and trace-optimization of polynomials. Math. Program. 137(1–2), 557–578 (2013)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Brešar, M., Klep, I.: Noncommutative polynomials, Lie skew-ideals and tracial Nullstellensätze. Math. Res. Lett. 16(4), 605–626 (2009)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Burgdorf, S., Klep, I.: The truncated tracial moment problem. J. Oper. Theory 68, 141–163 (2012)MathSciNetMATHGoogle Scholar
  6. 6.
    Brändén, P.: Obstructions to determinantal representability. Adv. Math. 226(2), 1202–1212 (2011)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Curto, R.E., Fialkow, L.A.: Solution of the truncated complex moment problem for flat data. Mem. Am. Math. Soc. 119(568), x+52 (1996)MathSciNetMATHGoogle Scholar
  8. 8.
    Curto, R.E., Fialkow, L.A.: Flat extensions of positive moment matrices: recursively generated relations. Mem. Am. Math. Soc. 136(648), x+56 (1998)MathSciNetMATHGoogle Scholar
  9. 9.
    Cimprič, J.: A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming. J. Math. Anal. Appl. 369(2), 443–452 (2010)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Cafuta, K., Klep, I., Povh, J.: Constrained polynomial optimization problems with noncommuting variables. SIAM J. Optim. 22(2), 363–383 (2012)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Connes, A.: Classification of injective factors. Cases \(II_{1}, II_{\infty }, III_{\lambda }, \lambda \ne 1\). Ann. Math. (2) 104(1), 73–115 (1976)CrossRefMathSciNetGoogle Scholar
  13. 13.
    de Klerk, E.: Aspects of Semidefinite Programming, Volume 65 of Applied Optimization. Kluwer, Dordrecht (2002)CrossRefGoogle Scholar
  14. 14.
    Helton, J.W., Klep, I., McCullough, S.: The convex Positivstellensatz in a free algebra. Adv. Math. 231(1), 516–534 (2012)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Henrion, D., Lasserre, J.B., Löfberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw., 24(4–5), 761–779 (2009). Available from
  16. 16.
    Helton, J.W., McCullough, S.: A Positivstellensatz for non-commutative polynomials. Trans. Am. Math. Soc. 356(9), 3721–3737 (2004)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Helton, J.W., McCullough, S.: Every convex free basic semi-algebraic set has an LMI representation. Ann. Math. (2) 176(2), 979–1013 (2012)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Helton, J.W., McCullough, S., de Oliveira, M.C., Putinar, M.: Engineering systems and free semi-algebraic geometry. In: Emerging Applications of Algebraic Geometry, Volume 149 of IMA Vol. Math. Appl., pp. 17–62. Springer, (2008)Google Scholar
  19. 19.
    Klep, I., Schweighofer, M.: Connes’ embedding conjecture and sums of Hermitian squares. Adv. Math. 217(4), 1816–1837 (2008)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Foundations of free noncommutative function theory, volume 199. American Mathematical Society, (2014)Google Scholar
  21. 21.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. J. Optim. 11(3), 796–817 (2000/2001)Google Scholar
  22. 22.
    Lasserre, J.B.: Moments, Positive Polynomials and Their Applications, vol. 1. Imperial College Press, London (2009)CrossRefGoogle Scholar
  23. 23.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging Applications of Algebraic Geometry, pp. 157–270. Springer, (2009)Google Scholar
  24. 24.
    Löfberg, J.: YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference, Taipei, Taiwan, 2004. Available from
  25. 25.
    Mittelmann, D.: An independent benchmarking of SDP and SOCP solvers. Math. Program. B, 95, 407–430 (2003).
  26. 26.
    Malick, J., Povh, J., Rendl, F., Wiegele, A.: Regularization methods for semidefinite programming. SIAM J. Optim. 20(1), 336–356 (2009)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Nie, J.: The \({\cal A}\)-truncated \({\cal K}\)-moment problem. Found. Comput. Math. 14(6), 1243–1276 (2014)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Netzer, T., Thom, A.: Hyperbolic polynomials and generalized Clifford algebras. Disc. Comput. Geom. 51, 802–814 (2014)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Pironio, S., Navascués, M., Acín, A.: Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM J. Optim. 20(5), 2157–2180 (2010)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Prajna, S., Papachristodoulou, A., Seiler, P., Parrilo, P.A.: SOSTOOLS and its control applications. In: Positive polynomials in control, Volume 312 of Lecture Notes in Control and Inform. Sci., pp. 273–292. Springer, Berlin, (2005)Google Scholar
  31. 31.
    Procesi, C.: The invariant theory of \(n\times n\) matrices. Adv. Math. 19(3), 306–381 (1976)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Procesi, C., Schacher, M.: A non-commutative real nullstellensatz and hilbert’s 17th problem. Ann. Math. 104(3), 395–406 (1976)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Powers, V., Scheiderer, C.: The moment problem for non-compact semialgebraic sets. Adv. Geom. 1(1), 71–88 (2001)CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Rowen, L.H.: Polynomial Identities in Ring Theory Volume 84 of Pure and Applied Mathematics, vol. 84. Academic Press Inc., New York (1980)Google Scholar
  36. 36.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw., 11/12(1–4), 625–653 (1999). Available from
  37. 37.
    Takesaki, M.: Theory of Operator Algebras. III. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
  38. 38.
    Toh, K.C., Todd, M.J., Tütüncü, R.H.: SDPT3—a MATLAB software package for semidefinite programming, version 1.3. Optim. Methods Softw., 11/12(1–4), 545–581 (1999). Available from
  39. 39.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)CrossRefMathSciNetMATHGoogle Scholar
  40. 40.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M., Sugimoto, H.: Algorithm 883: sparsePOP—a sparse semidefinite programming relaxation of polynomial optimization problems. ACM Trans. Math. Software, 35(2), Art. 15, 13 (2009)Google Scholar
  41. 41.
    Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming. Kluwer, Dordrecht (2000)CrossRefMATHGoogle Scholar
  42. 42.
    Xu, S., Lam, J.: A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci. 39(12), 1095–1113 (2008)CrossRefMathSciNetMATHGoogle Scholar
  43. 43.
    Yamashita, M., Fujisawa, K., Kojima, M.: Implementation and evaluation of SDPA 6.0 (semidefinite programming algorithm 6.0). Optim. Methods Softw., 18(4), 491–505 (2003). Available from

Copyright information

© Springer Science+Business Media New York (outside the USA) 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand
  2. 2.Faculty of information studies in Novo mestoNovo mestoSlovenia

Personalised recommendations