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Sparse noncommutative polynomial optimization

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Abstract

This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre (SIAM J Optim 17(3):822–843, 2006) and Waki et al. (SIAM J Optim 17(1):218–242, 2006). The Gelfand–Naimark–Segal construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue of noncommutative polynomials from the literature.

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References

  1. Anjos, M.F., Lasserre, J.B. (eds.): Handbook on Semidefinite, Conic and Polynomial Optimization, vol. 166. Springer, Berlin (2011)

    MATH  Google Scholar 

  2. Barvinok, A.: A course in convexity. Graduate Studies in Mathematics, vol. 54. American Mathematical Society, Providence, RI (2002)

  3. Burgdorf, S., Cafuta, K., Klep, I., Povh, J.: The tracial moment problem and trace-optimization of polynomials. Math. Program. 137(1–2, Ser. A), 557–578 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. Studies in Applied Mathematics, vol. 15. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  5. Burgdorf, S., Klep, I., Povh, J.: Optimization of Polynomials in Non-commuting Variables Springer Briefs in Mathematics. Springer, Cham (2016)

    MATH  Google Scholar 

  6. Blackadar, B.E.: Weak expectations and nuclear \(C^{\ast } \)-algebras. Indiana Univ. Math. J. 27(6), 1021–1026 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bessis, D., Moussa, P., Villani, M.: Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. J. Math. Phys. 16(11), 2318–2325 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berhuy, G., Oggier, F.: An Introduction to Central Simple Algebras and Their Applications to Wireless Communication Mathematical Surveys and Monographs, vol. 191. American Mathematical Society, Providence (2013)

    MATH  Google Scholar 

  9. Blair, J.R.S., Peyton, B.: An introduction to chordal graphs and clique trees. In: Graph Theory and Sparse Matrix Computation, volume 56 of IMA Volume Mathematics and its Applications, pp. 1–29. Springer, New York (1993)

  10. Bresar, M.: Introduction to Noncommutative Algebra. Universitext. Springer, Cham (2014)

    Book  MATH  Google Scholar 

  11. Conn, A.R., Gould, N.I.M., Toint, P.L.: Testing a class of methods for solving minimization problems with simple bounds on the variables. Math. Comput. 50(182), 399–430 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, T., Lasserre, J.B., Magron, V., Pauwels, E.: Semialgebraic optimization for lipschitz constants of ReLU networks. Adv. Neural Info. Process. Syst. 33 (2020)

  15. Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion. I. General framework. SIAM J. Optim. 11(3), 647–674 (2000/01)

  16. Gribling, S., de Laat, D., Laurent, M.: Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization. Math. Program. 170(1, Ser. B), 5–42 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gribling, S., de Laat, D., Laurent, M.: Lower bounds on matrix factorization ranks via noncommutative polynomial optimization. Found. Comput. Math. (2019) (to appear)

  18. Grimm, D., Netzer, T., Schweighofer, M.: A note on the representation of positive polynomials with structured sparsity. Arch. Math. 89(5), 399–403 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. William Helton, J., Klep, I., McCullough, S.: The convex Positivstellensatz in a free algebra. Adv. Math. 231(1), 516–534 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Henrion, D., Lasserre, J.-B., Löfberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Henrion, D., Lasserre, J.-B., Savorgnan, C.: Approximate volume and integration for basic semialgebraic sets. SIAM Rev. 51(4), 722–743 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Helton, J.W., McCullough, S.A.: A Positivstellensatz for non-commutative polynomials. Trans. Am. Math. Soc 356(9), 3721–3737 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jameson, G.: Ordered linear spaces. In: Ordered linear spaces, pp. 1–39. Springer (1970)

  25. Josz, C.: Application of polynomial optimization to electricity transmission networks. Université Pierre et Marie Curie - Paris VI, Theses (2016)

  26. Krivine, J.-L.: Anneaux préordonnés. J. Anal. Math. 12, 307–326 (1964)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Lam, T.-Y.: A First Course in Noncommutative Rings, vol. 131. Springer, Berlin (2013)

    Google Scholar 

  29. Lasserre, J.-B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2000/01)

  30. Lasserre, J.-B.: Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Laurent, M.: Matrix completion problems. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 1967–1975. Springer (2009)

  32. Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging Applications of Algebraic Geometry, volume 149 of The IMA Volumes in Mathematics and its Applications, pp. 157–270. Springer, New York (2009)

  33. Lax, P.D.: Differential equations, difference equations and matrix theory. Commun. Pure Appl. Math. 11, 175–194 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  34. Laurent, M., Rendl, F.: Semidefinite programming and integer programming. Handb. Oper. Res. Manag. Sci. 12, 393–514 (2005)

    MATH  Google Scholar 

  35. Lieb, E.H., Seiringer, R.: Equivalent forms of the Bessis–Moussa–Villani conjecture. J. Stat. Phys. 115(1–2), 185–190 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lasserre, J.-B., Toh, K.-C., Yang, S.: A bounded degree SOS hierarchy for polynomial optimization. EURO J. Comput. Optim. 5(1–2), 87–117 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Magron, V.: Interval enclosures of upper bounds of roundoff errors using semidefinite programming. ACM Trans. Math. Softw. 44(4), 41:1–41:18 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Magron, V., Constantinides, G., Donaldson, A.: Certified roundoff error bounds using semidefinite programming. ACM Trans. Math. Softw. 43(4), 1–31 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Murota, K., Kanno, Y., Kojima, M., Kojima, S.: A numerical algorithm for block-diagonal decomposition of matrix \(*\)-algebras with application to semidefinite programming. Jpn. J. Ind. Appl. Math. 27(1), 125–160 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mai, N.H.A., Lasserre, J.-B., Magron, V.: A sparse version of Reznick’s Positivstellensatz. arXiv preprint arXiv:2002.05101 (2020) (Submitted)

  42. The MOSEK optimization software. http://www.mosek.com/

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Nash, S.G.: Newton-type minimization via the Lánczos method. SIAM J. Numer. Anal. 21(4), 770–788 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  45. Nakata, K., Fujisawa, K., Fukuda, M., Kojima, M., Murota, K.: Exploiting sparsity in semidefinite programming via matrix completion. II. Implementation and numerical results. Math. Program. 95(2, Ser. B), 303–327 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Nie, J.: The \({\cal{A}}\)-truncated \(K\)-moment problem. Found. Comput. Math. 14(6), 1243–1276 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Navascués, M., Pironio, S., Acín, A.: A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys. 10(7), 073013 (2008)

    Article  Google Scholar 

  48. Netzer, T., Thom, A.: Hyperbolic polynomials and generalized Clifford algebras. Discrete Comput. Geom. 51(4), 802–814 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  50. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  51. Pál, Károly F., Vértesi, Tamás: Quantum bounds on Bell inequalities. Phys. Rev. A (3), 79(2), 022120, 12 (2009)

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

    Article  MathSciNet  MATH  Google Scholar 

  53. Riener, C., Theobald, T., Andrén, L.J., Lasserre, J.-B.: Exploiting symmetries in SDP-relaxations for polynomial optimization. Math. Oper. Res. 38(1), 122–141 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  54. Skelton, R.E., Iwasaki, T., Grigoriadis, K.M.: A unified algebraic approach to linear control design. The Taylor & Francis Systems and Control Book Series. Taylor & Francis, Ltd., London (1998)

  55. Stahl, H.R.: Proof of the BMV conjecture. Acta Math. 211(2), 255–290 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  56. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  57. Takesaki, M.: Theory of operator algebras. III, volume 127 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (2003). Operator Algebras and Non-commutative Geometry, 8

  58. Tütüncü, R.H., Toh, K.-C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. 95(2, Ser. B), 189–217 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  59. Tacchi, M., Weisser, T., Lasserre, J.-B., Henrion, D.: Exploiting sparsity for semi-algebraic set volume computation. preprint arXiv:1902.02976 (2019)

  60. Voiculescu, D.-V., Dykema, K.J., Nica, A.: Free random variables. CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992)

  61. Voiculescu, D.-V.: Symmetries of some reduced free product \(C^\ast \)-algebras. In: Operator Algebras and Their Connections with Topology and Ergodic Theory (Busteni, 1983), volume 1132 of Lecture Notes in Mathematics, pp. 556–588. Springer, Berlin (1985)

  62. Wittek, P.: Algorithm 950: Ncpol2sdpa-sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables. ACM Trans. Math. Softw. 41(3), 1–12 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  63. 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. Softw. 35(2), 1–13 (2009)

    Article  MathSciNet  Google Scholar 

  64. Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17(1), 218–242 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  65. Weisser, T., Lasserre, J.-B., Toh, K.-C.: Sparse-BSOS: a bounded degree SOS hierarchy for large scale polynomial optimization with sparsity. Math. Program. Comput. 10(1), 1–32 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  66. Wang, J., Magron, V.: Exploiting term sparsity in noncommutative polynomial optimization. arXiv preprint arXiv:2010.06956 (2020)

  67. Wang, J., Magron, V., Lasserre, J.-B.: TSSOS: a moment-SOS hierarchy that exploits term sparsity. arXiv preprint arXiv:1912.08899 (2019)

  68. Wang, J., Magron, V., Lasserre, J.-B.: Chordal-TSSOS: a moment-SOS hierarchy that exploits term sparsity with chordal extension. arXiv preprint arXiv:2003.03210 (2020)

  69. Wang, J., Magron, V., Lasserre, J.-B., Hoang A.M.: Ngoc: CS-TSSOS: correlative and term sparsity for large-scale polynomial optimization. arXiv preprint arXiv:2005.02828 (2020)

  70. Yamashita, M., Fujisawa, K., Kojima, M.: Implementation and evaluation of SDPA 6.0 (semidefinite programming algorithm 6.0). volume 18, pp. 491–505. 2003. The Second Japanese-Sino Optimization Meeting, Part II (Kyoto, 2002)

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Igor Klep

  2. LAAS-CNRS and Institute of Mathematics, Toulouse, France

    Victor Magron

  3. Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia

    Janez Povh

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  1. Igor Klep
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  2. Victor Magron
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  3. Janez Povh
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IK was supported by the Slovenian Research Agency Grants J1-2453, N1-0057 and P1-0222. Partially supported by the Marsden Fund Council of the Royal Society of New Zealand. VM was supported by the FMJH Program PGMO (EPICS project) and EDF, Thales, Orange et Criteo, as well as from the Tremplin ERC Stg Grant ANR-18-ERC2-0004-01 (T-COPS project). JP was supported bt the Slovenian Research Agency program P2-0256 and Grants J1-8132, J1-8155, N1-0057 and N1-0071.

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Klep, I., Magron, V. & Povh, J. Sparse noncommutative polynomial optimization. Math. Program. 193, 789–829 (2022). https://doi.org/10.1007/s10107-020-01610-1

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