Advertisement

Sums of Squares, Moment Matrices and Optimization Over Polynomials

  • Monique Laurent
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 149)

Abstract

We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

Key words

positive polynomial sum of squares of polynomials moment problem polynomial optimization semidefinite programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N.1. AKHIEZER. The classical moment problem, Hafner, New York, 1965.Google Scholar
  2. 2.
    C. BACHOC AND F. VALLENTIN, New upper bounds for kissing numbers from semidefinite programming, Journal of the American Mathematical Society, to appear.Google Scholar
  3. 3.
    --, Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps, arXiv:math.MG/0610856, 2006.Google Scholar
  4. 4.
    ,Optimality and uniqueness of the (4,10, 1/6)-sphericaJ code, arXiv:math.MG/0708.3947, 2007.Google Scholar
  5. 5.
    Y. BAI, E. DE KLERK, D.V. PASECHNIK, R. SOTIROV, Exploiting group symmetry in truss topology optimization, CentER Discussion paper 2007-17, Tilburg University, The Netherlands, 2007.Google Scholar
  6. 6.
    S. BASU, R. POLLACK, AND M.-F. RoY, Algorithms in Real Algebraic Geometry, Springer, 2003.Google Scholar
  7. 7.
    C. BAYER AND J. TEICHMANN, The proof of Tchakaloff's theorem, Proceedings of the American Mathematical Society 134:3035-3040, 2006.Google Scholar
  8. 8.
    [81 C. BERG. The multidimensional moment problem and semi-groups, Proc. Symp. Appl. Math. 37:110-124, 1987.Google Scholar
  9. 9.
    C. BERG, J.P.R. CHRISTENSEN, AND C.V. JENSEN, A remark on the multidimensional moment problem, Mathematische Annalen 243:163-169, 1979.Google Scholar
  10. 10.
    C. BERG, J.P.R. CHRISTENSEN, AND P. RESSEL, Positive definite Iunctions on Abelian semigroups, Mathematische Annalen 223:253-272, 1976.Google Scholar
  11. 11.
    C. BERG AND P.H. MASERICK, Exponentially bounded positive definite functions, Illinois Journal of Mathematics 28:162-179, 1984.Google Scholar
  12. 12.
    J.R.S. BLAIR AND B. PEYTON, An introduction to chordal graphs and clique trees, In Graph Theory and Sparse Matrix Completion, A. George, J.R. Gilbert, and J.W.H. Liu, eds, Springer-Verlag, New York, pp 1-29, 1993.Google Scholar
  13. 13.
    G. BLEKHERMAN, There are significantly more nonnegative polynomials than sums of squares, Isreal Journal of Mathematics, 153:355-380, 2006.Google Scholar
  14. 14.
    J. BOCHNAK, M. COSTE, AND M.-F. ROY, Geometrie Algebrique Reelle, Springer, Berlin, 1987. (Real algebraic geometry, second edition in english, Springer, Berlin, 1998.)Google Scholar
  15. 15.
    H.L. BODLAENDER AND K. JANSEN, On the complexity of the maximum cut problem, Nordic Journal of Computing 7(1):14-31, 2000.Google Scholar
  16. 16.
    H. BOSSE, Symmetric positive definite polynomials which are not sums ofsquares, preprint, 2007.Google Scholar
  17. 17.
    [17) M.-D. CROl AND T.-Y. LAM, Extremal positive semidefinite forms, Math. Ann. 231:1-18, 1977.Google Scholar
  18. 18.
    M.-D. CHOl, T.-Y. LAM, AND B. REZNICK, Sums of squares of real polynomials, Proceedings of Symposia in Pure mathematics 58(2):103-126, 1995.Google Scholar
  19. 19.
    A.M. COHEN, H. CUYPERS, AND H. STERK (EDS.), Some Tapas of Computer Algebra, Springer, Berlin, 1999.Google Scholar
  20. 20.
    R.M. CORLESS, P.M. GlANNI, AND B.M. TRAGER, A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots, Proceedings ACM International Symposium Symbolic and Algebraic Computations, Maui, Hawaii, 133-140, 1997.Google Scholar
  21. 21.
    D.A. Cox, J.B. LITTLE, AND D. O'SHEA, Ideals, Varieties, and Algotitibsus: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, 1997.Google Scholar
  22. 22.
    ---, Using Algebraic Geometry, Graduate Texts in Mathematics 185, Springer, New York, 1998.Google Scholar
  23. 23.
    R.E. CURTO AND L.A. FIALKOW, Solution of the truncated complex moment problem for flat data, Memoirs of the American Mathematical Society 119(568), 1996.Google Scholar
  24. 24.
    ---, Flat extensions of positive moment matrices: recursively generated relations, Memoirs of the American Mathematical Society, 136(648), 1998.Google Scholar
  25. 25.
    ---, The truncated complex K -moment problem Transactions of the American Mathematical Society 352:2825-2855, 2000.Google Scholar
  26. 26.
    ---, An analogue of the Riesz-Hevilend theorem for the truncated moment problem, preprint, 2007.Google Scholar
  27. 27.
    R.E. CURTO, L.A. FIALKOW, AND H.M. MOLLER, The extremal truncated moment problem, Integral Equations and Operator Theory, to appear.Google Scholar
  28. 28.
    E. DE KLERK, Aspects of Semidefinite Programming - Interior Point AlgorUhms and Selected Applications, Kluwer, 2002.Google Scholar
  29. 29.
    E. DE KLERK, D. DEN HERTOG, AND G. ELABWABI, Optimization of univariate functions on bounded intervals by interpolation and semidefinite programming, CentER Discussion paper 2006-26, Tilburg University, The Netherlands, April 2006.Google Scholar
  30. 30.
    E. DE KLERK, M. LAURENT, AND P. PARRILO, On the equivalence of algebraic approaches to the minimization of forms on the simplex, In Positive Polynomials in Control, D. HENRION AND A. GARULLl (EDS.), Lecture Notes on Control and Information Sciences 312:121-133, Springer, Berlin, 2005.Google Scholar
  31. 31.
    ---, A PTAS for the minimization of polynomials of fixed degree over the simplex, Theoretical Computer Science 361(2-3):210-225, 2006.Google Scholar
  32. 32.
    E. DE KLERK AND D.V. PASECHNIK, Approximating the stability number of a. graph via copositlve programming, SIAM Journal on Optimization 12:875892,2002.Google Scholar
  33. 33.
    E. DE KLERK, n.v. PASECHNIK, AND A. SCHRIJVER, Reduction of symmetric semidefinite programs using the regular *-representation, Mathematical Programming B 109: 613-624, 2007.Google Scholar
  34. 34.
    E. DE KLERK AND R. SOTlROV, Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem, Optimization Online, 2007.Google Scholar
  35. 35.
    J.L. KRrvlNE, Anneaux preordonnes, J. Anal. Math. 12:307-326, 1964.Google Scholar
  36. 36.
    ---, Quelques propn'etes des pteordtes dans les anneaux cominutetiis uiiiteires, C. R. Academic des Sciences de Paris 258:3417-3418, 1964.Google Scholar
  37. 37.
    A. DICKENSTEIN AND 1. Z. EMIRIS (EDS.). Solving Polynomial Equations: Foundations, Algorithms, and Applications, Algorithms and Computation in Mathematics 14, Springer-Verlag, 2005.Google Scholar
  38. 38.
    L.A. FIALKOW, Truncated multivariate moment problems with finite variety, Journal of Operator Theory, to appear.Google Scholar
  39. 39.
    B. FUGLEDE, The multidimensional moment problem, Expositiones Mathematicae 1:47---65,1983.Google Scholar
  40. 40.
    M. R. GAREY AND D.S. JOHNSON, Computers and IntractabiHty: A Guide to the Theory of NP-Completeness, San Francisco, W.H. Freeman & Company, Publishers, 1979.Google Scholar
  41. 41.
    K. GATERMAN AND P. PARRILO, Symmetry groups, semidefinite programs and sums of squares, Journal of Pure and Applied Algebra 192:9~128, 2004.Google Scholar
  42. 42.
    M.X. GOEMANS AND D. WILLIAMSON, Improved approximation algorithms for maximum cuts and satisfiabjJjty problems using semidefinite programming, Journal of the ACM 42:1115-1145, 1995.Google Scholar
  43. 43.
    D. GRIMM, T. NETZER, AND M. SCHWEIGHOFER, A note on the representation of positive polynomials with structured sparsity, Archiv der Mathematik, to appear.Google Scholar
  44. 44.
    B. GRONE, C.R. JOHNSON, E. MARQUES DE SA, AND H. WOLKOWICZ, Positive definite completions of partial Hermitian matrices, Linear Algebra and its Applications 58:109-124, 1984.Google Scholar
  45. 45.
    M. GROTSCHEL, L. LOVASZ, AND A. SCHRIJVER, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988.Google Scholar
  46. 46.
    N. GVOZDENOVIC AND M. LAURENT, Semidefinite bounds for the stabjlity number of a graph via sums of squares of polynomials, Mathematical Programming 110(1):145-173, 2007.Google Scholar
  47. 47.
    ~-~, The operator III for the cbrouietic number of a graph, SIAM Journal on Optimization, to appear.Google Scholar
  48. 48.
    ---, Computing semidefinite programming lower bounds for the (fractional) chromatic number via block-diagonalization, SIAM Journal on Optimization, to appear.Google Scholar
  49. 49.
    D. HANDELMAN, Representing polynomials by positive linear functions on compact convex polyhedra, Pacific Journal of Mathematics 132(1):35----52, 1988.Google Scholar
  50. 50.
    B. HANZON AND D. JIBETEAN, Global minimization of a mulUvariate polynomial using matrix methods, Journal of Global Optimization 27:1-23, 2003.Google Scholar
  51. 51.
    E.K. HAVILAND, On the momentum problem for distributions in more than one dimension, American Journal of Mathematics 57:562-568, 1935.Google Scholar
  52. 52.
    J.W. HELTON AND M. PUTINAR, Positive polynomials in scalar and matrix variables, the spectral theorem and optimization. In Operator Theory, Structured Matrices, and Dilations: Tiberiu Constantinescu Memorial Volume, M. Bakonyi, A. Gheondea, M. Putinar, and J. Rovnyak (eds.), Theta, Bucharest, pp. 229-306, 2007.Google Scholar
  53. 53.
    D. HENRION AND J.-B. LASSERRE, GlopUPoly: Global optimization over polynomials with Matlab and SeDuMi, ACM Transactions Math. Soft. 29:165-194, 2003.Google Scholar
  54. 54.
    --, Detecting glo bal optimality and extracting solutions in GloptiPoly, In Positive Polynomials in Control, D. HENRION AND A. GARULLI (EDS.), Lecture Notes on Control and Information Sciences 312:293-310, Springer, Berlin, 2005.Google Scholar
  55. 55.
    D. HENRION, J.-B. LA SSERRE, AND J. LOFBERG, GloptiPoly 3: moments, optimization and semidefinite programming, arXiv:0709.2559, 2007.Google Scholar
  56. 56.
    D. HILBERT, Uber die Darstellung deiiniter Formen als Summe von Formenquadraten, Mathematische Annalen 32:342-350, 1888. See Ges. Abh. 2:154161, Springer, Berlin, reprinted by Chelsea, New York, 1981.Google Scholar
  57. 57.
    T. JACOBI AND A. PRESTEL, Distinguished representations of strictly positive polynomials, Journal fur die Reine und Angewandte Mathematik 532:223235, 200l.Google Scholar
  58. 58.
    L. JANSSON, J.B. LASSERRE, C. RIENER, AND T. THEOBALD, Exploiting symmetries in SDP-relaxations for polynomial optimization, Optimization Online, 2006.Google Scholar
  59. 59.
    D. JIBETEAN, Algebraic Optimization with Applications to System Theory, Ph.D thesis, Vrije Universiteit, Amsterdam, The Netherlands, 2003.Google Scholar
  60. 60.
    D. JIBETEAN AND M. LAURENT, Semidefinite epproxittuulotis for global unconstrained polynomial optimization, SIAM Journal on Optimization 16:490514, 2005.Google Scholar
  61. 61.
    M. KOJIMA, S. KIM, AND H. WAKI, Sparsity in sums of squares of polynomials, Mathematical Programming 103:45--62,2005.Google Scholar
  62. 62.
    M. KOJIMA AND M. MURAMATSU, A note on sparse SOS and SDP relaxatl"ons for polynomial optimization problems over symmetric cones, Research Report B421, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2006.Google Scholar
  63. 63.
    S. KUHLMAN AND M. MARSHALL, Positivity, sums of squares and the multidimensional moment problem, Transactions of the American Mathematical Society 354:4285~4301, 2002.Google Scholar
  64. 64.
    H. LANDAU (ed.), Moments in Mathematics, Proceedings of Symposia in Applied Mathematics 37, 1-15, AMS, Providence, 1987.Google Scholar
  65. 65.
    J. B. LASSERRE, Global optimization with polynomials and the problem of moments, SIAM Journal on Optimization 11:796-817, 200l.Google Scholar
  66. 66.
    ---, An explicit exact SDP relaxation for nonlinear 0 - 1 programs In K. AARDAL AND A.M.H. GERARDS (EDS.), Lecture Notes in Computer Science 2081:293-303, 200l.Google Scholar
  67. 67.
    ---, Polynomials nonnegative on a grid and discrete representations, Transactions of the American Mathematical Society 354:631--649, 200l.Google Scholar
  68. 68.
    ---, Semidefinite programming vs LPrelaxations for polynomial programming, Mathematics of Operations Research 27:347-360, 2002.Google Scholar
  69. 69.
    -~-, SOS approximations of polynoiuiels nonnegative on a real algebraic set, SIAM Journal on Optimization 16:610--628, 2005.Google Scholar
  70. 70.
    ---, Polynomial programming: LP-relaxations also converge, SIAM Journal on Optimization 15:383-393~ 2005.Google Scholar
  71. 71.
    ---, A sum of squares approximation of nonnegaUve polynomials, SIAM Journal on Optimization 16:751-765, 2006.Google Scholar
  72. 72.
    ---, Convergent semidefinite relaxations in polynomial optimizeiion with sparsity, SIAM Journal on Optimization 17:822-843, 2006.Google Scholar
  73. 73.
    ---, Sufficient conditions for a real polynomial to be a sum of squares, Archiv der Mathematik, to appear.Google Scholar
  74. 74.
    ---, A PosUivstellensatz which preserves the coupling pattern of variables, Preprint.Google Scholar
  75. 75.
    J. B. LASSERRE, M. LAURENT, AND P. ROSTALSKI, Semidefinite characterization and computation of real radical ideals, Foundations of Computational Mathematics, to appear.Google Scholar
  76. 76.
    ---, A unified approach for real and complex zeros of zero-dimensional ideals, this volume, IMA Volumes in Mathematics and its Applications, Emerging Applications of Algebraic Geometry, M. Putinar and S. Sullivant, eds., Springer Science-l-Business Media, New York, 2008.Google Scholar
  77. 77.
    J.B. LASSERRE AND T. NETZER, SOS approximations of nonnegative polynomials via simple high degree perturbations, Mathematische Zeitschrift 256:99-112, 2006.Google Scholar
  78. 78.
    M. LAURENT, A comparison of the Sherali-Adams, Lovesz-Scbrijver and Lasserre relaxations for 0-1 programming, Mathematics of Operations Research 28(3):470-496, 2003.Google Scholar
  79. 79.
    ---, Semidefinite relaxations for Max-Cut, In The Sharpest Cut: The Impact of Manfred Padberg and His Work. M. Grotschel, ed., pp. 257-290, MPSSIAM Series in Optimization 4, 2004.Google Scholar
  80. 80.
    ---, Revisiting two theorems of Curto and Fialkow on moment matrices, Proceedings of the American Mathematical Society 133(10):2965~2976,2005.Google Scholar
  81. 81.
    ---, Semidefinite representations for finite varieties. Mathematical Programming, 109:1-26, 2007.Google Scholar
  82. 82.
    [82J -~-, Strengthened semidefinite programming bounds for codes, Mathematical Programming B 109:239-261, 2007.Google Scholar
  83. 83.
    M. LAURENT AND F. RENDL, Semidefinite Programming and Integer Programming, In Handbook on Discrete Optimization, K. Aardal, G. Nemhauser, R. Weismantel (eds.), pp. 393-514, Elsevier B.V., 2005.Google Scholar
  84. 84.
    J. LOFBERG AND P. PARRILO, From coefficients to samples: a new approach to SOS optimization, 43rd IEEE Conference on Decision and Control, Vol. 3, pp. 3154-3159, 2004.Google Scholar
  85. 85.
    [851 L. LOVASZ, On the Shannon capacity of a graph. IEEE Transactions on Information Theory IT-25:1-7, 1979.Google Scholar
  86. 86.
    [861 L. LovASZ AND A. SCHRlJVER, Cones of matrices and set-functions and 0-1 optimization, SIAM Journal on Optimization 1:166-190, 1991.Google Scholar
  87. 87.
    M. MARSHALL, Positive polynomials and sums of squares, Dottorato de Ricerca in Matematica, Dipartimento di Matematica dell'Universita di Pisa, 2000.Google Scholar
  88. 88.
    ---, Optimization of polynomial functions, Canadian Math. Bull. 46:575-587, 2003.Google Scholar
  89. 89.
    ---, Representation of non-negative polynomials, degree bounds and applications to optimization, Canad. J. Math., to appear.Google Scholar
  90. 90.
    ---, Positive Polynomials and Sums of Squares, AMS Surveys and Monographs, forthcoming book.Google Scholar
  91. 91.
    H.M. MOLLER, An inverse problem for cubature formulae, Vychislitel'nye Tekhnologii (Computational Technologies) 9:13-20, 2004.Google Scholar
  92. 92.
    H.M. MOLLER AND H.J. STETTER, Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems, Numerische Mathematik 70:311329, 1995.Google Scholar
  93. 93.
    T.S. MOTZK1N AND E.G. STRAUS, Maxima for graphs and a new proof of a theorem of Turan, Canadian Journal of Mathematics 17:533-540, 1965.Google Scholar
  94. 94.
    K.G. MURTY AND S.N. KABAD1, Some NP-complete problems in quadratic and nonlinear programming, Mathematical Programming 39:117-129, 1987.Google Scholar
  95. 95.
    Y.E. NESTEROV, Squared functional systems and optimization problems, In J.B.G. FRENK, C. Roos, T. TERLAKY, AND S. ZHANG (EDS.), High Performance Optimization, 405-440, Kluwer Academic Publishers, 2000.Google Scholar
  96. 96.
    Y.E. NESTEROV AND A. NEMJROVSK1, Interior Point Polynomial Methods in Convex Programming, Studies in Applied Mathematics, vol. 13, SIAM, Philadelphia, PA, 1994.Google Scholar
  97. 97.
    J. NIE, Sum of squares method for sensor network localization, Optimization Online, 2006Google Scholar
  98. 98.
    J. NlE AND J. DEMMEL, Sparse SOS Relaxations for Minimizing Functions that are Summation of Small Polynomials, ArXiv math.OC/0606476, 2006.Google Scholar
  99. 99.
    J. NIE, J. DEMMEL, AND B. STURMFELS, Minimizing polynomials via sums of squares over the gradient ideal, Mathematical Programming Series A 106:587-606, 2006.Google Scholar
  100. 100.
    J. N1E AND M. SCHWEIGHOFER, On the complexity of Putitier's Positivstellensatz, Journal of Complexity 23(1):135-150, 2007.Google Scholar
  101. 101.
    J. NOCEDAL AND 8.J. WRIGHT, Numerical Optimization, Springer Verlag, 2000.Google Scholar
  102. 102.
    [102J A.E. NUSSBAUM, Quasi-analytic vectors, Archiv. Mat. 6:179-191, 1966.Google Scholar
  103. 103.
    [103J P.A. PARR1LO, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, PhD thesis, California Institute of Technology, 2000.Google Scholar
  104. 104.
    ~~, Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming B 96:293-320, 2003.Google Scholar
  105. 105.
    --, An explicit construction of distinguished representations of polynomials nonnegative over finite sets, IfA Technical Report AUT02-02, ETH Zurich, 2002.Google Scholar
  106. 106.
    --, Exploiting algebraic structure in sum of squares programs, In Positive Polynomials in Control, D. Henrion and A. Garulli, eds., LNCIS 312:181194, 2005.Google Scholar
  107. 107.
    P.A. PARR1LO AND B. STURMFELS, Minimizing polynomial functions, In Algorithmic and Quantitative Real Algebraic geometry, S. Basu and L. GonzaJesVega, eds., DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 60, pp. 83-99, 2003.Google Scholar
  108. 108.
    G. POLYA, Uber positive Darstellung von Polynomen, Vierteljahresschrift der Naturforschenden Gesellschaft in Zurich 73:141-145, 1928. Reprinted in Collected Papers, vol. 2, MIT Press, Cambridge, MA, pp. 309-313, 1974.Google Scholar
  109. 109.
    V. POWERS AND B. REZNICK, A new bound for P61ya '5 theorem wUh applications to polynomials positive on polyhedra, Journal of Pure and Applied Algebra 164(1-2):221-229, 200l.Google Scholar
  110. 110.
    V. POWERS, B. REZNICK, C. SCHEIDERER, AND F. SO'ITILE, A new approach to Hilbert'« theorem on ternary quartics, C. R. Acad. Sci. Paris, Ser. I 339:617620, 2004.Google Scholar
  111. 111.
    V. POWERS AND C. SCHEJDERER, The moment problem for non-compact semialgebraic sets, Adv. Geom. 1:71-88, 2001.Google Scholar
  112. 112.
    (112} V. POWERS AND T. WORMANN, An algorithm for sums of squares of real polynomials, Journal of Pure and Applied Algebra 127:99-104, 1998.Google Scholar
  113. 113.
    S. PRAJNA, A. PAPACHRISTODOULOU, P. SEILER, AND P. A. PARR1LO, SOSTOOLS (Sums of squares optimization toolbox for MATLAB) User's guide, http://wvw.cds.caltech.edu/sostools/
  114. 114.
    A. PRESTEL AND C.N. DELZELL, Positive Polynomials - From Hilbert's 17 th Problem to Real Algebra, Springer, Berlin, 200l.Google Scholar
  115. 115.
    M. PUTINAR, Positive polynomials on compact semi-algebraic sets, Indiana University Mathematics Journal 42:969-984, 1993.Google Scholar
  116. 116.
    --, A note on Tchakaloff's theorem, Proceedings of the American Mathematical Society 125:2409-2414, 1997.Google Scholar
  117. 117.
    J. RENEGAR, A Mathematical View of Interior-Point Methods in Convex Optimization, MPS-SIAM Series in Optimization, 200l.Google Scholar
  118. 118.
    B. REZNICK, Extremal PSD forms with few terms, Duke Mathematical Journal 45(2):363-374, 1978.Google Scholar
  119. 119.
    ---, Uniform denominators in Hilbert's Seventeenth Problem, Mathematische Zeitschrift 220: 75-98, 1995.Google Scholar
  120. 120.
    ---, Some concrete aspects of Hilbert's 17th problem, In Real Algebraic Geometry and Ordered Structures, C.N. DELZELL AND J.J. MADDEN (EDS.), Contemporary Mathematics 253:251-272, 2000.Google Scholar
  121. 121.
    ---, On Hilbert's construction of positive polynomials, 2007, http://front. matb.ucdavis.edu/0707.2156.
  122. 122.
    T. ROH AND L. VANDENBERGHE, Discrete transforms, semidefinite programming and sum-oi-squares representations of nonnegative polynomials, SIAM Journal on Optimization 16:939-964, 2006.Google Scholar
  123. 123.
    J. SAXE, EmbeddabjJity of weighted graphs in k-spece is strongly NP-hard, In Proc, 17th Allerton Conf. in Communications, Control, and Computing, Monticello, IL, pp. 480-489, 1979.Google Scholar
  124. 124.
    C. SCHE1DERER, Personal communication, 2004.Google Scholar
  125. 125.
    --Positjvity and sums of squares: A guide to recent results, this volume, IMA Volumes in Mathematics and its Applications, Emerging Applications of Algebraic Geometry, M. Putinar and S. Sullivant, eds., Springer Science- l-Business Media, New York, 2008.Google Scholar
  126. 126.
    K. SCHMUDGEN, The K-moment problem for compact semi-algebraic sets, Mathematische Annalen 289:203-206, 1991.Google Scholar
  127. 127.
    ---Noncommutative real algebraic geometry - Some basic concepts and first ideas, this volume, IMA Volumes in Mathematics and its Applications, Emerging Applications of Algebraic Geometry, M. Putinar and S. Sullivant, eds., Springer Sclence-l-Business Media, New York, 2008.Google Scholar
  128. 128.
    A. SCHRIJVER, Theory of Linear and Integer Programming, Wiley, 1979.Google Scholar
  129. 129.
    ---, New code upper bounds from the Terwilliger algebra and semidetuiite programming, IEEE Trans. Inform. Theory 51:2859-2866, 2005.Google Scholar
  130. 130.
    M. SCHWEIGHOFER, An algorithmic approach to Schmiidgen's Positivstellensatz, Journal of Pure and Applied Algebra, 166:307-319, 2002.Google Scholar
  131. 131.
    ---, On the complexity of Schmiidgen's Positivstellensatz, Journal of Complexity 20:529-543, 2004.Google Scholar
  132. 132.
    ---, Optimization of polynomials on compact semialgebraic sets, SIAM Journal on Optimization 15(3):805-825, 2005.Google Scholar
  133. 133.
    ---, Global optimization of polynomials using gradient tentacles and sums of squares, SIAM Journal on Optimization 17(3):920-942, 2006.Google Scholar
  134. 134.
    ---, A Grabner basis proof of the flat extension theorem for moment matrices, Preprint, 2008.Google Scholar
  135. 135.
    J. -P. SERRE, Linear Representation of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer Verlag, New York, 1977.Google Scholar
  136. 136.
    LR. SHAFAREVICH, Basic Algebraic Geometry, Springer, Berlin, 1994.Google Scholar
  137. 137.
    H.D. SHERALI AND W.P. ADAMS, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM Journal on Discrete Mathematics, 3:411-430, 1990.Google Scholar
  138. 138.
    N.Z. SHOR, An approach to obtaining global extremums in polynomial mathematical programming problems, Kibernetika, 5:102-106, 1987.Google Scholar
  139. 139.
    ---, Class of global minimum bounds of polynomial functions, Cybernetics 23(6):731-734, 1987. (Russian orig.: Kibernetika 6:9-11, 1987.)Google Scholar
  140. 140.
    -~-, Quadratic optimization problems, Soviet J. Comput. Systems Sci. 25:111, 1987.Google Scholar
  141. 141.
    ---, Nondifferentiable Optimization and Polynomial Problems, Kluwer, Dordrecht, 1998.Google Scholar
  142. 142.
    G. STENGLE, A Nullstellensatz and a Positivstellenseiz in semialgebrajc geometry, Math. Ann. 207~87-97, 1974.Google Scholar
  143. 143.
    J. STOCHEL, Solving the truncated moment problem solves the moment problem, Glasgow Journal of Mathematics, 43:335-341, 2001.Google Scholar
  144. 144.
    B. STURMFELS, Solving Systems of Polynomial Equations. CBMS, Regional Conference Series in Mathematics, Number 97, AMS, Providence, 2002.Google Scholar
  145. 145.
    V. TCHAKALOFF, Formules de cubatures mecaniques a coefficients non negatifs. Bulletin des Sciences Mathematiques, 81: 123-134, 1957.Google Scholar
  146. 146.
    J. A. TELLE AND A. PROSKUROWSK1, Algorithms for vertex pertitlonitig problems on partial k-trees, SIAM Journal on Discrete Mathematics 10(4):529-550, 1997.Google Scholar
  147. 147.
    M. TODD, Semidefinite optimization, Acata Numer. 10:515-560, 2001.Google Scholar
  148. 148.
    F. VALLENTIN, Optimal Embeddings of Distance Regular Graphs into Euclidean Spaces, Journal of Combinatorial Theory, Series B 98:95-104, 2008.Google Scholar
  149. 149.
    ---, Symmetry in semidefinite programs, arXiv:0706.4233, 2007.Google Scholar
  150. 150.
    L. VANDENBERGHE AND S. BOYD, Semidefinite programming, SIAM Review 38:49-95, 1996.Google Scholar
  151. 151.
    L. VAN DEN DRIES, Tame topology and o-minimal structures, Cambridge University Press, Cambridge, 1998.Google Scholar
  152. 152.
    H. WAKI, Personal communication, 2007.Google Scholar
  153. 153.
    H. WAKI, S. KIM, M. KOJIMA, AND M. MURAMATSU, Sums ofsquares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity, SIAM Journal on Optimization 17(1):218-242, 2006.Google Scholar
  154. 154.
    H. WHITNEY, Elementary structure of real algebraic varieties, Annals of Mathematics 66(3):545-556, 1957.Google Scholar
  155. 155.
    T. WIMER, Linear Time Algorithms on k- Terminal Graphs, Ph.D. thesis, Clemson University, Clemson, se, 1988.Google Scholar
  156. 156.
    H. WOLKOWICZ, R. SAIGAL, AND L. VANDEBERGHE (eds.), Handbook of Semidefinite Programming, Boston, Kluwer Academic, 2000.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Monique Laurent
    • 1
  1. 1.CWIAmsterdamThe Netherlands

Personalised recommendations