Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Symmetries in Polynomial Optimization

  • Chapter
  • First Online:
Polynomial Optimization, Moments, and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 206))

  • 493 Accesses

Abstract

This chapter investigates how symmetries can be used to reduce the computational complexity in polynomial optimization problems. A focus will be specifically given on the Moment-SOS hierarchy in polynomial optimization, where results from representation theory and invariant theory of groups can be used. In addition, symmetry reduction techniques which are more generally applicable are also presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Acevedo, J., Velasco, M.: Test sets for nonnegativity of polynomials invariant under a finite reflection group. J. Pure Appl. Algebra 220(8), 2936–2947 (2016)

    MathSciNet  Google Scholar 

  2. Ariki, S., Terasoma, T., Yamada, H.-F.: Higher Specht polynomials. Hiroshima Math. J. 27(1), 177–188 (1997)

    MathSciNet  Google Scholar 

  3. Bachoc, C., Vallentin, F.: New upper bounds for kissing numbers from semidefinite programming. J. Am. Math. Soc. 21(3), 909–924 (2008)

    MathSciNet  Google Scholar 

  4. Bachoc, C., Gijswijt, D.C., Schrijver, A., Vallentin, F.: Invariant semidefinite programs. In: Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 219–269. Springer, Berlin (2012)

    Google Scholar 

  5. Basu, S., Riener, C.: Bounding the equivariant Betti numbers of symmetric semi-algebraic sets. Adv. Math. 305, 803–855 (2017)

    MathSciNet  Google Scholar 

  6. Basu, S., Riener, C.: Efficient algorithms for computing the Euler-Poincaré characteristic of symmetric semi-algebraic sets. In: Ordered Algebraic Structures and Related Topics: International Conference on Ordered Algebraic Structures and Related Topics, October 12–16, 2015, Centre International de Rencontres Mathématiques (CIRM), Luminy, vol. 697, pp. 53–81. American Mathematical Society, Providence (2017)

    Google Scholar 

  7. Basu, S., Riener, C.: On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets: polynomial bounds on multiplicities. Int. Math. Res. Not. 2020(7), 2054–2113 (2020)

    MathSciNet  Google Scholar 

  8. Basu, S., Riener, C.: Vandermonde varieties, mirrored spaces, and the cohomology of symmetric semi-algebraic sets. Found. Comput. Math. 22(5), 1395–1462 (2022)

    MathSciNet  Google Scholar 

  9. Benson, D.J.: Polynomial Invariants of Finite Groups, vol. 190. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  10. Bergeron, F.: Algebraic Combinatorics and Coinvariant Spaces. CRC Press, Boca Raton (2009)

    Google Scholar 

  11. Bien, F.: Constructions of telephone networks by group representations. Not. Am. Math. Soc. 36, 5–22 (1989)

    MathSciNet  Google Scholar 

  12. Blekherman, G.: Nonnegative polynomials and sums of squares. J. Am. Math. Soc. 25(3), 617–635 (2012)

    MathSciNet  Google Scholar 

  13. Blekherman, G., Riener, C.: Symmetric non-negative forms and sums of squares. Discret. Comput. Geom. 65(3), 764–799 (2021)

    MathSciNet  Google Scholar 

  14. Bröcker, L.: On symmetric semialgebraic sets and orbit spaces. Banach Center Publ. 44(1), 37–50 (1998)

    MathSciNet  Google Scholar 

  15. Chandrasekaran, V., Shah, P.: Relative entropy relaxations for signomial optimization. SIAM J. Optim. 26(2), 1147–1173 (2016)

    MathSciNet  Google Scholar 

  16. Cimprič, J., Kuhlmann, S., Scheiderer, C.: Sums of squares and moment problems in equivariant situations. Trans. Am. Math. Soc. 361(2), 735–765 (2009)

    MathSciNet  Google Scholar 

  17. de Laat, D., Vallentin, F.: A semidefinite programming hierarchy for packing problems in discrete geometry. Math. Program. 151(2), 529–553 (2015)

    MathSciNet  Google Scholar 

  18. de Laat, D., Keizer, W.d.M., Machado, F.C.: The Lasserre hierarchy for equiangular lines with a fixed angle (2022). arXiv preprint arXiv:2211.16471

    Google Scholar 

  19. Debus, S., Riener, C.: Reflection groups and cones of sums of squares. J. Symb. Comput. 119, 112–144 (2023)

    MathSciNet  Google Scholar 

  20. Derksen, H., Kemper, G.: Computational Invariant Theory. Springer, Berlin (2015)

    Google Scholar 

  21. Dostert, M., Guzmán, C., de Oliveira Filho, F.M., Vallentin, F.: New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry. Discrete Comput. Geom. 58(2), 449–481 (2017)

    MathSciNet  Google Scholar 

  22. Faugère, J.-C., Labahn, G., El Din, M.S., Schost, É., Vu, T.X.: Computing critical points for invariant algebraic systems. J. Symb. Comput. 116, 365–399 (2023)

    MathSciNet  Google Scholar 

  23. Friedl, T., Riener, C., Sanyal, R.: Reflection groups, reflection arrangements, and invariant real varieties. Proc. Am. Math. Soc. 146(3), 1031–1045 (2018)

    MathSciNet  Google Scholar 

  24. Fulton, W., Harris, J.: Representation Theory: A First Course, vol. 129. Springer Science & Business Media, Berlin (2013)

    Google Scholar 

  25. Garey, M., Johnson, D., Stockmeyer, L.: Some simplified np-complete problems. In: Proc. ACM Symp. on Theory of Computing, pp. 47–63 (1974)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  27. Gijswijt, D., Schrijver, A., Tanaka, H.: New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming. J. Comb. Theory A 113(8), 1719–1731 (2006)

    MathSciNet  Google Scholar 

  28. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)

    MathSciNet  Google Scholar 

  29. Görlach, P., Riener, C., Weißer, T.: Deciding positivity of multisymmetric polynomials. J. Symb. Comput. 74, 603–616 (2016)

    MathSciNet  Google Scholar 

  30. Hannabuss, K.: Sound and symmetry. Math. Intell. 19(4), 16–20 (1997)

    MathSciNet  Google Scholar 

  31. Haviland, E.: On the momentum problem for distribution functions in more than one dimension. II. Am. J. Math. 58(1), 164–168 (1936)

    MathSciNet  Google Scholar 

  32. Heaton, A., Shankar, I.: An sos counterexample to an inequality of symmetric functions. J. Pure Appl. Algebra 225(8), 106656 (2021)

    MathSciNet  Google Scholar 

  33. Herr, K., Rehn, T., Schürmann, A.: Exploiting symmetry in integer convex optimization using core points. Oper. Res. Lett. 41(3), 298–304 (2013)

    MathSciNet  Google Scholar 

  34. Hilbert, D.: Über die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann. 32(3), 342–350 (1888)

    MathSciNet  Google Scholar 

  35. Hubert, E., Bazan, E.R.: Algorithms for fundamental invariants and equivariants. Math. Comput. 91(337), 2459–2488 (2022)

    MathSciNet  Google Scholar 

  36. Hubert, E., Metzlaff, T., Moustrou, P., Riener, C.: Optimization of trigonometric polynomials with crystallographic symmetry and applications (2022). HAL preprint hal-03768067

    Google Scholar 

  37. Humphreys, J.E.: Reflection Groups and Coxeter Groups, vol. 29. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  38. James, G.D.: The Representation Theory of the Symmetric Groups, vol. 682. Springer, Berlin (2006)

    Google Scholar 

  39. Judd, B.R.: Lie groups in atomic and molecular spectroscopy. SIAM J. Appl. Math. 25(2), 186–192 (1973)

    MathSciNet  Google Scholar 

  40. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Springer, Berlin (1972)

    Google Scholar 

  41. Kurpisz, A., Leppänen, S., Mastrolilli, M.: Sum-of-squares hierarchy lower bounds for symmetric formulations. Math. Program. 182(1), 369–397 (2020)

    MathSciNet  Google Scholar 

  42. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    MathSciNet  Google Scholar 

  43. Lasserre, J.-B.: Moments, Positive Polynomials and Their Applications, vol. 1. World Scientific, Singapore (2009)

    Google Scholar 

  44. Laurent, M.: Strengthened semidefinite programming bounds for codes. Math. Program. 109(2), 239–261 (2007)

    MathSciNet  Google Scholar 

  45. Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging Applications of Algebraic Geometry, vol. 149. IMA Vol. Math. Appl., pp. 157–270. Springer, New York (2009)

    Google Scholar 

  46. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25(1), 1–7 (1979)

    MathSciNet  Google Scholar 

  47. Mourrain, B., Vrahatis, M.N., Yakhoubson, J.-C.: On the complexity of isolating real roots and computing with certainty the topological degree. J. Complexity 182, 612–640 (2002)

    MathSciNet  Google Scholar 

  48. Moustrou, P., Riener, C., Verdure, H.: Symmetric ideals, specht polynomials and solutions to symmetric systems of equations. J. Symb. Comput. 107, 106–121 (2021)

    MathSciNet  Google Scholar 

  49. Moustrou, P., Naumann, H., Riener, C., Theobald, T., Verdure, H.: Symmetry reduction in am/gm-based optimization. SIAM J. Optim. 32(2), 765–785 (2022)

    MathSciNet  Google Scholar 

  50. Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39, 117–129 (1987)

    MathSciNet  Google Scholar 

  51. Nagata, M.: On the 14-th problem of Hilbert. Am. J. Math. 81(3), 766–772 (1959)

    MathSciNet  Google Scholar 

  52. Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)

    Google Scholar 

  53. Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, Massachusetts Institute of Technology (2000)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  55. Procesi, C., Schwarz, G.: Inequalities defining orbit spaces. Invent. Math. 81(3), 539–554 (1985)

    MathSciNet  Google Scholar 

  56. Ramana, M.V.: An exact duality theory for semidefinite programming and its complexity implications. Math. Program. 77(1), 129–162 (1997)

    MathSciNet  Google Scholar 

  57. Raymond, A., Saunderson, J., Singh, M., Thomas, R.R.: Symmetric sums of squares over k-subset hypercubes. Math. Program. 167(2), 315–354 (2018)

    MathSciNet  Google Scholar 

  58. Riener, C.: Symmetries in semidefinite and polynomial optimization: relaxations, combinatorics, and the degree principle. PhD thesis, Johann Wolfgang Goethe-Universität Frankfurt am Main (2011)

    Google Scholar 

  59. Riener, C.: On the degree and half-degree principle for symmetric polynomials. J. Pure Appl. Algebra 216(4), 850–856 (2012)

    MathSciNet  Google Scholar 

  60. Riener, C.: Symmetric semi-algebraic sets and non-negativity of symmetric polynomials. J. Pure Appl. Algebra 220(8), 2809–2815 (2016)

    MathSciNet  Google Scholar 

  61. Riener, C., Schabert, R.: Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions (2022). arXiv preprint arXiv:2203.08727

    Google Scholar 

  62. 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)

    MathSciNet  Google Scholar 

  63. 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)

    MathSciNet  Google Scholar 

  64. Schürmann, A.: Exploiting symmetries in polyhedral computations. Fields Inst. Commun. 69, 265–278 (2013)

    MathSciNet  Google Scholar 

  65. Serre, J.-P., et al.: Linear Representations of Finite Groups, vol. 42. Springer, Berlin (1977)

    Google Scholar 

  66. Specht, W.: Zur Darstellungstheorie der symmetrischen Gruppe. Math. Z. 42(1), 774–779 (1937)

    MathSciNet  Google Scholar 

  67. Springer, T.A.: Invariant Theory, vol. 585. Springer, Berlin (2006)

    Google Scholar 

  68. Stanley, R.P.: Enumerative Combinatorics Volume 1, 2nd edn. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2011)

    Google Scholar 

  69. Steinberg, R.: Invariants of finite reflection groups. Can. J. Math. 12, 616–618 (1960)

    MathSciNet  Google Scholar 

  70. Stiefel, E., Fässler, A.: Group Theoretical Methods and Their Applications. Springer Science & Business Media, Berlin (2012)

    Google Scholar 

  71. Sturmfels, B.: Algorithms in Invariant Theory. Springer Science & Business Media, Berlin (2008)

    Google Scholar 

  72. Timofte, V.: On the positivity of symmetric polynomial functions.: Part i: General results. J. Math. Anal. Appl. 284(1), 174–190 (2003)

    Google Scholar 

  73. Turski, J.: Geometric fourier analysis of the conformal camera for active vision. SIAM Rev. 46(2), 230–255 (2004)

    MathSciNet  Google Scholar 

  74. Vallentin, F.: Symmetry in semidefinite programs. Linear Algebra Appl. 430(1), 360–369 (2009)

    MathSciNet  Google Scholar 

  75. Vu, T.X.: Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials. In: ISSAC ’2012–Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, pp. 167–175 (2022)

    Google Scholar 

  76. Waterhouse, W.C.: Do symmetric problems have symmetric solutions? Am. Math. Mon. 90(6), 378–387 (1983)

    MathSciNet  Google Scholar 

  77. Wigner, E.P.: Reduction of direct products and restriction of representations to subgroups: the everyday tasks of the quantum theorists. SIAM J. Appl. Math. 25(2), 169–185 (1973)

    MathSciNet  Google Scholar 

  78. Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, vol. 27. Springer Science & Business Media, Berlin (2012)

    Google Scholar 

Download references

Acknowledgements

This expository article originates from a minicourse “Symmetries in algorithmic questions in real algebraic geometry” held during the virtual POEMA (Polynomial Optimization, Efficiency Through Moments and Algebra) Learning Weeks in July 2020. The authors would like to thank the anonymous reviewers for helpful comments.

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Toulouse Jean Jaurès, Toulouse, France

    Philippe Moustrou

  2. Institutt for Matematikk og Statistikk, UiT - Norges arktiske universitet, Tromsø, Norway

    Cordian Riener & Hugues Verdure

Authors
  1. Philippe Moustrou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cordian Riener
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hugues Verdure
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cordian Riener .

Editor information

Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Michal Kočvara

  2. Aromath, Inria at Université Côte d’Azur, Sophia Antipolis, France

    Bernard Mourrain

  3. UiT The Arctic University of Norway, Tromsø, Norway

    Cordian Riener

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Moustrou, P., Riener, C., Verdure, H. (2023). Symmetries in Polynomial Optimization. In: Kočvara, M., Mourrain, B., Riener, C. (eds) Polynomial Optimization, Moments, and Applications. Springer Optimization and Its Applications, vol 206. Springer, Cham. https://doi.org/10.1007/978-3-031-38659-6_3

Download citation

Publish with us

Policies and ethics

Search

Navigation