Invariant Semidefinite Programs

  • Christine Bachoc
  • Dion C. Gijswijt
  • Alexander Schrijver
  • Frank Vallentin
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 166)

Abstract

This chapter provides the reader with the necessary background for dealing with semidefinite programs which have symmetry. The basic theory is given and it is illustrated in applications from coding theory, combinatorics, geometry, and polynomial optimization.

References

  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. Cambridge University Press, Cambridge (1999)Google Scholar
  2. 2.
    Astola, J.: The Lee-scheme and bounds for Lee-codes. Cybernet. Systems 13, 331–343 (1982)Google Scholar
  3. 3.
    Bachoc, C.: Linear programming bounds for codes in Grassmannian spaces. IEEE Trans. Inf. Th. 52, 2111–2125 (2006)Google Scholar
  4. 4.
    Bachoc, C.: Semidefinite programming, harmonic analysis and coding theory. arXiv:0909.4767 [cs.IT] (2009)Google Scholar
  5. 5.
    Bachoc, C., Vallentin, F.: New upper bounds for kissing numbers from semidefinite programming. J. Amer. Math. Soc. 21, 909–924 (2008)Google Scholar
  6. 6.
    Bachoc, C., Vallentin, F.: Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps, special issue in the honor of Eichii Bannai, Europ. J. Comb. 30 625–637 (2009)Google Scholar
  7. 7.
    Bachoc, C., Vallentin, F.: More semidefinite programming bounds (extended abstract). pages 129–132 in Proceedings “DMHF 2007: COE Conference on the Development of Dynamic Mathematics with High Functionality”, October 2007, Fukuoka, Japan. (2007)Google Scholar
  8. 8.
    Bachoc, C., Vallentin, F.: Optimality and uniqueness of the (4,10,1/6) spherical code. J. Comb. Theory Ser. A 116, 195–204 (2009)Google Scholar
  9. 9.
    Bachoc, C., Zémor, G.: Bounds for binary codes relative to pseudo-distances of k points. Adv. Math. Commun. 4, 547–565 (2010)Google Scholar
  10. 10.
    Bachoc, C., Nebe G., de Oliveira Filho, F.M., Vallentin, F.: Lower bounds for measurable chromatic numbers. Geom. Funct. Anal. 19,645–661 (2009)Google Scholar
  11. 11.
    Bai, Y., de Klerk, E., Pasechnik, D.V., Sotirov, R.: Exploiting group symmetry in truss topology optimization. Optimization and Engineering 10, 331–349 (2009)Google Scholar
  12. 12.
    Bannai, E., Ito, T.: Algebraic combinatorics. I.. The Benjamin/Cummings Publishing Co. Inc., Menlo Park, CA (1984)Google Scholar
  13. 13.
    Barg, A., Purkayastha, P.: Bounds on ordered codes and orthogonal arrays. Moscow Math. Journal 9, 211–243 (2009)Google Scholar
  14. 14.
    Barvinok, A.: A course in convexity. Graduate Studies in Mathematics 54, American Mathematical Society (2002)Google Scholar
  15. 15.
    Berger, M.: A Panoramic View of Riemannian Geometry. Springer-Verlag (2003)Google Scholar
  16. 16.
    Bochner, S.: Hilbert distances and positive definite functions. Ann. of Math. 42, 647–656 (1941)Google Scholar
  17. 17.
    Bosse, H.: Symmetric, positive polynomials, which are not sums of squares. Series: CWI. Probability, Networks and Algorithms [PNA], Nr. E0706 (2007)Google Scholar
  18. 18.
    Boyd, S., Diaconis, P., Parrilo, P.A., Xiao, L.: Symmetry analysis of reversible Markov chains. Internet Mathematics 2, (2005)Google Scholar
  19. 19.
    Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-regular graphs. Springer-Verlag, Berlin (1989)Google Scholar
  20. 20.
    Bump, D.: Lie Groups. Graduate Text in Mathematics 225, Springer-Verlag (2004)Google Scholar
  21. 21.
    Cameron, P. J.: Coherent configurations, association schemes and permutation groups. pages 55–71 in Groups, combinatorics & geometry (Durham, 2001), World Sci. Publishing, River Edge, NJ (2003)Google Scholar
  22. 22.
    Cimpric̆, J.: A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming. J. Math. Anal. Appl. 369, 443–452 (2010)Google Scholar
  23. 23.
    Cimpric̆, J., Kuhlmann, S., Scheiderer, C.: Sums of squares and moment problems in equivariant situations. Trans. Amer. Math. Soc. 361, 735–765 (2009).Google Scholar
  24. 24.
    Conway, J.B.: A course in functional analysis. Graduate Text in Mathematics 96, Springer-Verlag (2007)Google Scholar
  25. 25.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. third edition, Springer-Verlag, New York (1999)Google Scholar
  26. 26.
    Creignou, J., Diet, H.: Linear programming bounds for unitary codes. Adv. Math. Commun. 4, 323–344 (2010)Google Scholar
  27. 27.
    Davidson, K.R.: C*-Algebras by Example. Fields Institute Monographs 6, American Mathematical Society (1996)Google Scholar
  28. 28.
    Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. (1973)Google Scholar
  29. 29.
    Delsarte, P., Goethals, J. M.: Alternating bilinear forms over GF(q). J. Comb. Th. A 19, 26–50 (1975)Google Scholar
  30. 30.
    Delsarte, P.: Hahn polynomials, discrete harmonics and t-designs. SIAM J. Appl. Math. 34, 157–166 (1978)Google Scholar
  31. 31.
    Delsarte, P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Th. A 25, 226–241 (1978)Google Scholar
  32. 32.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata 6, 363–388 (1977)Google Scholar
  33. 33.
    de Klerk, E., Dobre, C., Pasechnik, D.V.: Numerical block diagonalization of matrix *-algebras with application to semidefinite programming. To appear in Math. Program., Ser. B. (2009)Google Scholar
  34. 34.
    Duffin, R.J.: Infinite Programs. In: H.W. Kuhn, A.W. Tucker (eds.) Linear inequalities and related systems, Princeton Univ. Press, 157–170 (1956)Google Scholar
  35. 35.
    Dunkl, C.F.: A Krawtchouk polynomial addition theorem and wreath product of symmetric groups. Indiana Univ. Math. J. 25, 335–358 (1976)Google Scholar
  36. 36.
    Ericson, T., Zinoviev, V.: Codes on Euclidean spheres. North-Holland (2001)Google Scholar
  37. 37.
    Gatermann, K., Parrilo, P.A.: Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Alg. 192, 95–128 (2004)Google Scholar
  38. 38.
    Gijswijt, D.C.: Matrix Algebras and Semidefinite Programming Techniques for Codes. Dissertation, University of Amsterdam (2005)Google Scholar
  39. 39.
    Gijswijt, D.C.: Block diagonalization for algebra’s associated with block codes. arXiv:0910.4515 [math.OC] (2009)Google Scholar
  40. 40.
    Gijswijt, D.C., Mittelmann, H.D., Schrijver, A.: Semidefinite code bounds based on quadruple distances. arXiv.math:1005.4959 [math.CO] (2010)Google Scholar
  41. 41.
    Gijswijt, D.C., Schrijver, A., Tanaka, H.: New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming. J. Comb. Theory Ser. A 113, 1719–1731 (2006)Google Scholar
  42. 42.
    Goemans, M.X., Williamson, D.P.: Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming. J. Comput. System Sci. 68, 442–470 (2004)Google Scholar
  43. 43.
    Gvozdenović, N.: Approximating the stability number and the chromatic number of a graph via semidefinite programming. Dissertation, University of Amsterdam (2008)Google Scholar
  44. 44.
    Gvozdenović, N., Laurent, M.: The operator Ψ for the chromatic number of a graph. SIAM J. Optim. 19, 572–591 (2008)Google Scholar
  45. 45.
    Gvozdenović, N., Laurent, M.: Computing semidefinite programming lower bounds for the (fractional) chromatic number via block-diagonalization. SIAM J. Optim. 19, 592–615 (2008)Google Scholar
  46. 46.
    Gvozdenović, N., Laurent, M., Vallentin, F.: Block-diagonal semidefinite programming hierarchies for 0/1 programming. Oper. Res. Lett. 37, 27–31 (2009)Google Scholar
  47. 47.
    Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, Cambridge (1990)Google Scholar
  48. 48.
    James, A.T., Constantine, A.G.: Generalized Jacobi polynomials as spherical functions of the Grassmann manifold. Proc. London Math. Soc. 29, 174–192 (1974)Google Scholar
  49. 49.
    Jansson, L., Lasserre, J.B., Riener, C., Theobald, T.: Exploiting symmetries in SDP-relaxations for polynomial optimization. Optimization Online, September 2006, (2006)Google Scholar
  50. 50.
    Kabatiansky, G.A., Levenshtein, V.I.: Bounds for packings on a sphere and in space. Problems of Information Transmission 14, 1–17 (1978)Google Scholar
  51. 51.
    Kanno, Y., Ohsaki, M., Murota, K., Katoh, N.: Group symmetry in interior-point methods for semidefinite program. Optimization and Engineering 2, 293–320 (2001)Google Scholar
  52. 52.
    Kleitman, D.J.: The crossing number of K 5, n. J. Comb. Theory Ser. B 9, 315–323 (1970)Google Scholar
  53. 53.
    de Klerk, E.: Exploiting special structure in semidefinite programming: A survey of theory and applications. European Journal of Operational Research 201, 1–10 (2010)Google Scholar
  54. 54.
    de Klerk, E., Maharry, J., Pasechnik, D.V., Richter, R.B., Salazar, G.: Improved bounds for the crossing numbers of K m, n and K n. SIAM J. Disc. Math. 20, 189–202 (2006)Google Scholar
  55. 55.
    de Klerk, E., Newman, M.W., Pasechnik, D.V., Sotirov R.: On the Lovasz θ-number of almost regular graphs with application to Erdős-Rényi graphs. European J. Combin. 31, 879–888 (2009)Google Scholar
  56. 56.
    de Klerk, E., Pasechnik, D.V.: Solving SDP’s in non-commutative algebras part I: the dual-scaling algorithm. Discussion paper from Tilburg University, Center for economic research (2005)Google Scholar
  57. 57.
    de Klerk, E., Pasechnik, D.V.: A note on the stability number of an orthogonality graph. European J. Combin. 28, 1971–1979 (2007)Google Scholar
  58. 58.
    de Klerk, E., Pasechnik, D.V., Schrijver, A.: Reduction of symmetric semidefinite programs using the regular ∗ -representation. Math. Program., Ser. B 109, 613–624 (2007)Google Scholar
  59. 59.
    de Klerk, E., Pasechnik, D.V., Sotirov, R.: On Semidefinite Programming Relaxations of the Travelling Salesman Problem. Discussion paper from Tilburg University, Center for economic research (2007)Google Scholar
  60. 60.
    de Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. Math. Program. Ser. A 122, 225–246 (2010)Google Scholar
  61. 61.
    Knuth, D.E.: The sandwich theorem. Electron. J. Combin. 1, (1994)Google Scholar
  62. 62.
    Lasserre, J.B.: An explicit exact SDP relaxation for nonlinear 0/1 programs. In: K. Aardal and A.M.H. Gerards (eds.) Lecture Notes in Computer Science 2081 (2001)Google Scholar
  63. 63.
    Laurent, M.: Strengthened semidefinite programming bounds for codes. Math. Program. Ser. B 109, 239–261 (2007)Google Scholar
  64. 64.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: M. Putinar, S. Sullivant (eds.) Emerging Applications of Algebraic Geometry, Vol. 149 of IMA Volumes in Mathematics and its Applications, Springer-Verlag (2009)Google Scholar
  65. 65.
    Levenshtein, V.I.: Universal bounds for codes and designs. In: V. Pless, W. C. Huffmann (eds.) Handbook of Coding Theory, Elsevier, Amsterdam (1998)Google Scholar
  66. 66.
    Linial, N., Magen, A., Naor, A.: Girth and Euclidean Distortion. Geom. Funct. Anal. 12 380–394 (2002)Google Scholar
  67. 67.
    Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25, 1–5 (1979)Google Scholar
  68. 68.
    MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford (1977)Google Scholar
  69. 69.
    Martin, W.J., Stinson, D.R.: Association schemes for ordered orthogonal arrays and (T, M, S)-nets. Canad. J. Math. 51, 326–346 (1999)Google Scholar
  70. 70.
    McEliece, R.J., Rodemich, E.R., Rumsey, Jr, H.C.: The Lovász bound and some generalizations. J. Combinatorics, Inform. Syst. Sci. 3, 134–152 (1978)Google Scholar
  71. 71.
    Mittelmann, H.D., Vallentin, F.: High accuracy semidefinite programming bounds for kissing numbers. Experiment. Math. 19, 174–178 (2010)Google Scholar
  72. 72.
    Murota, K., Kanno, Y., Kojima, M., Kojima, S.: A numerical algorithm for block-diagonal decomposition of matrix *-algebras, Part I: proposed approach and application to semidefinite programming. Jpn. J. Ind. Appl. Math. 27, 125–160 (2010)Google Scholar
  73. 73.
    Musin, O.R.: Multivariate positive definite functions on spheres. arXiv:math/0701083 [math.MG] (2007)Google Scholar
  74. 74.
    Nemirovski, A.: Advances in convex optimization: Conic programming. In: M. Sanz-Sol, J. Soria, J.L. Varona, J. Verdera, (eds.) Proceedings of International Congress of Mathematicians, Madrid, August 22-30, 2006, Volume 1, European Mathematical Society Publishing House (2007)Google Scholar
  75. 75.
    Odlyzko, A.M., Sloane, N.J.A.: New bounds on the number of unit spheres that can touch a unit sphere in n dimensions. J. Comb. Theory Ser. A 26, 210–214 (1979)Google Scholar
  76. 76.
    de Oliveira Filho, F.M.: New Bounds for Geometric Packing and Coloring via Harmonic Analysis and Optimization. Dissertation, University of Amsterdam (2009)Google Scholar
  77. 77.
    de Oliveira Filho, F.M., Vallentin, F.: Fourier analysis, linear programming, and densities of distance avoiding sets in \({\mathbb{R}}^{n}\). J. Eur. Math. Soc. (JEMS) 12, 1417–1428 (2010)Google Scholar
  78. 78.
    Pasechnik, D.V., Kini, K.: A GAP package for computation with coherent configurations. In: K. Fukuda, J. van der Hoeven, M. Joswig, N. Takayama (eds.) Mathematical Software — ICMS 2010 LNCS 6327, Springer (2010)Google Scholar
  79. 79.
    Roy, A., Scott, A. J.: Unitary designs and codes. Des. Codes Cryptogr. 53, 13–31 (2009)Google Scholar
  80. 80.
    Roy, A.: Bounds for codes and designs in complex subspaces. arXiv:0806.2317 [math.CO] (2008)Google Scholar
  81. 81.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill International Editions (1987)Google Scholar
  82. 82.
    Schoenberg, I.J.: Positive definite functions on spheres. Duke Math. J. 9, 96–108 (1942)Google Scholar
  83. 83.
    Schrijver, A.: Association schemes and the Shannon capacity: Eberlein polynomials and the Erdős-Ko-Rado theorem. Algebraic methods in graph theory Vol. I, II (Szeged, 1978), pp. 671–688, Colloq. Math. Soc. János Bolyai 25, North-Holland, Amsterdam-New York (1981)Google Scholar
  84. 84.
    Schrijver, A.: New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Trans. Inform. Theory 51, 2859–2866 (2005)Google Scholar
  85. 85.
    Soifer, A.: The mathematical coloring book. Springer-Verlag (2008)Google Scholar
  86. 86.
    Stanton, D.: Orthogonal polynomials and Chevalley groups. In: R.A. Askey, T.H. Koornwinder, W. Schempp (eds.) Special functions: group theoretical aspects and applications, Reidel Publishing Compagny (1984)Google Scholar
  87. 87.
    Sturm, J.F.: Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software 11, 625–653 (1999)Google Scholar
  88. 88.
    Sturmfels, B.: Algorithms in Invariant Theory. Springer-Verlag (1993)Google Scholar
  89. 89.
    Szegö, G.: Orthogonal polynomials. American Mathematical Society (1939)Google Scholar
  90. 90.
    Takesaki, M.: Theory of Operator Algebras I. Encyclopaedia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, 5. Springer-Verlag, Berlin, (2002)Google Scholar
  91. 91.
    Tarnanen, H., Aaltonen, M., Goethals J.M.: On the nonbinary Johnson scheme. European J. Combin. 6, 279–285 (1985)Google Scholar
  92. 92.
    Tarnanen, H.: Upper bounds on permutation codes via linear programming. European J. Combin. 20, 101–114 (1999)Google Scholar
  93. 93.
    Vallentin, F.: Optimal embeddings of distance transitive graphs into Euclidean spaces. J. Comb. Theory Ser. B 98, 95–104 (2008)Google Scholar
  94. 94.
    Vallentin, F.: Symmetry in semidefinite programs. Linear Algebra and Appl. 430, 360–369 (2009)Google Scholar
  95. 95.
    Vallentin, F.: Lecture notes: Semidefinite programs and harmonic analysis. arXiv:0809.2017 [math.OC] (2008)Google Scholar
  96. 96.
    Vilenkin, N.Ja., Klimyk, A.U.: Representation of Lie Groups and Special Functions, Volume 2. Kluwer Academic Publishers (1993)Google Scholar
  97. 97.
    Wang, H.G.: Two-point homogeneous spaces. Ann. of Math. 55, 177–191 (1952)Google Scholar
  98. 98.
    Woodall, D.R.: Cyclic-order graphs and Zarankiewicz’s crossing-number conjecture. J. Graph Theory 17, 657–671 (1993)Google Scholar
  99. 99.
    Zarankiewicz, K.: On a problem of P. Turán concerning graphs. Fundamenta Mathematicae 41, 137–145 (1954)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Christine Bachoc
    • 1
  • Dion C. Gijswijt
    • 2
    • 3
  • Alexander Schrijver
    • 4
    • 3
  • Frank Vallentin
    • 5
  1. 1.Institut de Mathématiques de BordeauxUniversité Bordeaux ITalenceFrance
  2. 2.CWI and Department of MathematicsLeiden UniversityLeidenThe Netherlands
  3. 3.Centrum voor Wiskunde en Informatica (CWI)AmsterdamThe Netherlands
  4. 4.CWI and Department of MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  5. 5.Delft Institute of Applied MathematicsTechnical University of DelftDelftThe Netherlands

Personalised recommendations