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

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Abstract

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.

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References

  • 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 York

    Book  Google Scholar 

  • 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 219–269

  • Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. Studies in applied mathematics. SIAM, Philadelphia

    Book  Google Scholar 

  • Burgdorf S, Cafuta K, Klep I, Povh J (2013a) Algorithmic aspects of sums of hermitian squares of noncommutative polynomials. Comput Optim Appl 55(1):137–153

    Article  Google Scholar 

  • Burgdorf S, Cafuta K, Klep I, Povh J (2013b) The tracial moment problem and trace-optimization of polynomials. Math Program 137(1):557–578

    Article  Google Scholar 

  • Burgdorf S, Klep I, Povh J (2016) Optimization of polynomials in non-commuting variables. Springer briefs in mathematics. Springer, Berlin

    Google Scholar 

  • Cafuta K (2013) On matrix algebras associated to sum-of-squares semidefinite programs. Linear Multilinear Algebra 61(11):1496–1509

    Article  Google Scholar 

  • 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, 10

    Article  Google Scholar 

  • 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. http://ncsostools.fis.unm.si/

  • Cafuta K, Klep I, Povh J (2012) Constrained polynomial optimization problems with noncommuting variables. SIAM J Optim 22(2):363–383

    Article  Google Scholar 

  • Cafuta K, Klep I, Povh J (2015) Rational sums of hermitian squares of free noncommutative polynomials. Ars Math Contemp 9(2):243–259

    Article  Google Scholar 

  • 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 103–126

  • Cimprič J (2010) A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming. J Math Anal Appl 369(2):443–452

    Article  Google Scholar 

  • 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 17–61

  • Gatermann K, Parrilo P (2004) Symmetry groups, semidefinite programs, and sums of squares. J Pure Appl Algebra 192(1–3):95–128

    Article  Google Scholar 

  • Goemans MX (1997) Semidefinite programming in combinatorial optimization. Math Program 79(1–3, Ser. B):143–161

    Google Scholar 

  • Halická M, de Klerk E, Roos C (2002) On the convergence of the central path in semidefinite optimization. SIAM J Optim 12(4):1090–1099

    Article  Google Scholar 

  • Helton J (2002) “Positive” noncommutative polynomials are sums of squares. Ann of Math 156(2):675–694

    Article  Google Scholar 

  • Helton J, Nie J (2012) A semidefinite approach for truncated K-moment problems. Found Comput Math 12(6):851–881

    Article  Google Scholar 

  • Horn R, Johnson C (1985) Matrix analysis. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Klep I, Povh J (2010) Semidefinite programming and sums of hermitian squares of noncommutative polynomials. J Pure Appl Algebra 214:740–749

    Article  Google Scholar 

  • Klep I, Povh J (2016) Constrained trace-optimization of polynomials in freely noncommuting variables. J Global Optim 64(2):325–348

    Article  Google Scholar 

  • Klep I, Schweighofer M (2008a) Connes’ embedding conjecture and sums of Hermitian squares. Adv Math 217(4):1816–1837

    Article  Google Scholar 

  • Klep I, Schweighofer M (2008b) Sums of Hermitian squares and the BMV conjecture. J Stat Phys 133(4):739–760

    Article  Google Scholar 

  • Lasserre J (2000/2001) Global optimization with polynomials and the problem of moments. SIAM J Optim 11(3):796–817

  • Lasserre J (2009) Moments, positive polynomials and their applications, vol 1. Imperial college press optimization series. Imperial College Press, London

    Google Scholar 

  • 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 157–270

  • Marshall M (2008) Positive polynomials and sums of squares, mathematical surveys and monographs, vol 146. American Mathematical Society, Providence

    Book  Google Scholar 

  • McCullough S (2001) Factorization of operator-valued polynomials in several non-commuting variables. Linear Algebra Appl 326(1–3):193–203

    Article  Google Scholar 

  • McCullough S, Putinar M (2005) Noncommutative sums of squares. Pac J Math 218(1):167–171

    Article  Google Scholar 

  • Mittelman H http://plato.asu.edu/sub/pns.html

  • Nie J (2009) Sum of squares method for sensor network localization. Comput Optim Appl 43(2):151–179

    Article  Google Scholar 

  • Nie J (2014) Optimality conditions and finite convergence of Lasserre’s hierarchy. Math Program 146(1–2):97–121

    Article  Google Scholar 

  • Parrilo P (2000) Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD. thesis, California Institute of Technology

  • Parrilo P (2003) Semidefinite programming relaxations for semialgebraic problems. Math Program 96(2, Ser. B):293–320

    Article  Google Scholar 

  • Pironio S, Navascués M, Acín A (2010) Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM J Optim 20(5):2157–2180

    Article  Google Scholar 

  • Powers V, Wörmann T (1998) An algorithm for sums of squares of real polynomials. J Pure Appl Algebra 127(1):99–104

    Article  Google Scholar 

  • Recht B, Fazel M, Parrilo P (2010) Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev 52(3):471–501

    Article  Google Scholar 

  • Reznick B (1978) Extremal PSD forms with few terms. Duke Math J 45(2):363–374

    Article  Google Scholar 

  • Shor NZ (1991) Dual quadratic estimates in polynomial and boolean programming. Ann Oper Res 25(1–4):163–168

    Google Scholar 

  • Sturm J (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11/12(1–4):625–653. http://sedumi.ie.lehigh.edu/

  • 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. http://www.math.nus.edu.sg/~mattohkc/sdpt3.html

  • Wolkowicz H, Saigal R, Vandenberghe L (2000) Handbook of semidefinite programming. Kluwer, Dordrecht

    Book  Google Scholar 

  • 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. http://sdpa.sourceforge.net/

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Acknowledgements

The author thanks both anonymous referees for helpful suggestions.

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Authors and Affiliations

  1. Laboratorij za uporabno matematiko in statistiko, Fakulteta za elektrotehniko, Univerza v Ljubljani, Tržaška 25, 1000, Ljubljana, Slovenia

    Kristijan Cafuta

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  1. Kristijan Cafuta
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Correspondence to Kristijan Cafuta.

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The author acknowledge the financial support from the Slovenian Research Agency (Research Core Funding No. P1-0222 and Project J1-8132).

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Cafuta, K. Sums of Hermitian squares decomposition of non-commutative polynomials in non-symmetric variables using NCSOStools. Cent Eur J Oper Res 27, 397–413 (2019). https://doi.org/10.1007/s10100-018-0533-z

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  • DOI: https://doi.org/10.1007/s10100-018-0533-z

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