Mathematical Methods of Operations Research

, Volume 78, Issue 1, pp 35–59 | Cite as

Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem

  • Alexander Engau
  • Miguel F. Anjos
  • Immanuel Bomze
Original Article

Abstract

The stable-set problem is an NP-hard problem that arises in numerous areas such as social networking, electrical engineering, environmental forest planning, bioinformatics clustering and prediction, and computational chemistry. While some relaxations provide high-quality bounds, they result in very large and expensive conic optimization problems. We describe and test an integrated interior-point cutting-plane method that efficiently handles the large number of nonnegativity constraints in the popular doubly-nonnegative relaxation. This algorithm identifies relevant inequalities dynamically and selectively adds new constraints in a build-up fashion. We present computational results showing the significant benefits of this approach in comparison to a standard interior-point cutting-plane method.

Keywords

Stable set Maximum clique Theta number  Semidefinite programming Interior-point algorithms  Cutting-plane methods Combinatorial optimization 

Mathematical Subject Classifications

90C09 90C20 90C22 90C27 90C35 90C51 90C90 

References

  1. Alizadeh F (1995) Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J Optim 5(1):13–51MathSciNetMATHCrossRefGoogle Scholar
  2. Anjos MF, Vannelli A (2008) Computing globally optimal solutions for single-row layout problems using semidefinite programming and cutting planes. INFORMS J Comput 20(4):611–617MathSciNetMATHCrossRefGoogle Scholar
  3. Balas E, Ceria S, Cornuéjols G (1993) A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math Program 58(3, Ser. A):295–324Google Scholar
  4. Barahona F, Epstein R, Weintraub A (1992) Habitat dispersion in forest planning and the stable set problem. Oper Res 40(1):S14–S21CrossRefGoogle Scholar
  5. Benson HY, Shanno DF (2007) An exact primal-dual penalty method approach to warmstarting interior-point methods for linear programming. Comput Optim Appl 38(3):371–399MathSciNetMATHCrossRefGoogle Scholar
  6. Bomze IM (2012) Copositive optimization-recent developments and applications. Eur J Oper Res 216(3):509–520MathSciNetMATHCrossRefGoogle Scholar
  7. Bomze IM, De Klerk E (2002) Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J Global Optim 24(2):163–185MathSciNetMATHCrossRefGoogle Scholar
  8. Bomze IM, Dür M, Teo CP (2012) Copositive optimization. Optima MOS Newsl 89:2–10. http://www.mathopt.org/Optima-Issues/optima89.pdf
  9. Bomze IM, Frommlet F, Locatelli M (2010) Copositivity cuts for improving SDP bounds on the clique number. Math Program 124(1–2, Ser. B):13–32Google Scholar
  10. Bomze IM, Locatelli M, Tardella F (2008) New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability. Math Program 115(1, Ser. A):31–64Google Scholar
  11. Bomze IM, Schachinger W, Uchida G (2012) Think co(mpletely)positive! Matrix properties, examples and a clustered bibliography on copositive optimization. J Global Optim 52(3):423–445MathSciNetMATHCrossRefGoogle Scholar
  12. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeMATHGoogle Scholar
  13. Burer S (2010) Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math Program Comput 2(1):1–19MathSciNetMATHCrossRefGoogle Scholar
  14. Burer S (2011) Copositive programming. In: Anjos MF, Lasserre JB (eds) Handbook of semidefinite, cone and polynomial optimization: theory, algorithms, software and applications. International Series in Operations Research and Management Science. Springer, New York, pp 201–218Google Scholar
  15. Dantzig GB, Ye Y (1991) A build-up interior method for linear programming: affine scaling form. Technical report, Stanford University/The University of IowaGoogle Scholar
  16. Davi T, Jarre F (2011) Solving large scale problems over the doubly nonnegative cone. Technical report, Institut für Informatik, Universität Düsseldorf (submitted). http://www.optimization-online.org/DB_HTML/2011/04/3000.html
  17. de Klerk E (2002) Aspects of semidefinite programming, applied optimization. Interior point algorithms and selected applications, vol 65. Kluwer, DordrechtGoogle Scholar
  18. de Klerk E (2010) Exploiting special structure in semidefinite programming: a survey of theory and applications. Eur J Oper Res 201:1–10MATHCrossRefGoogle Scholar
  19. de Klerk E, Pasechnik DV (2002) Approximation of the stability number of a graph via copositive programming. SIAM J Optim 12(4):875–892MathSciNetMATHCrossRefGoogle Scholar
  20. den Hertog D, Roos C, Terlaky T (1994) Adding and deleting constraints in the logarithmic barrier method for LP. In: Advances in optimization and approximation, vol 1. Nonconvex Optim Appl. Kluwer, Dordrecht, pp 166–185Google Scholar
  21. Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91(2, Ser. A):201–213Google Scholar
  22. Dukanovic I, Rendl F (2007) Semidefinite programming relaxations for graph coloring and maximal clique problems. Math Program 109(2–3, Ser. B):345–365Google Scholar
  23. Dukanovic I, Rendl F (2010) Copositive programming motivated bounds on the stability and the chromatic numbers. Math Program 121(2, Ser. A):249–268Google Scholar
  24. Dür M (2010) Copositive programming—a survey. In: Moritz D, Francois G, Elias J, Wim M (eds) Recent advances in optimization and its applications in engineering. Springer, Berlin, pp 3–20Google Scholar
  25. El-Bakry AS, Tapia RA, Zhang Y (1994) A study of indicators for identifying zero variables in interior-point methods. SIAM Rev 36(1):45–72MathSciNetMATHCrossRefGoogle Scholar
  26. Engau A (2012) Recent progress in interior-point methods: cutting plane methods and warm starts. In: Anjos MF, Lasserre JB (eds) Handbook of semidefinite, conic, and polynomial optimization, International Series in Operations Research & Management Science, Chapter 17, vol 166. Springer, New York, pp 471–498Google Scholar
  27. Engau A, Anjos MF (2011) A primal-dual interior-point algorithm for linear programming with selective addition of inequalities. Technical Report G-2011-44, GERAD, Montréal, QC, CanadaGoogle Scholar
  28. Engau A, Anjos MF, Vannelli A (2009) A primal-dual slack approach to warmstarting interior-point methods for linear programming. In: Chinneck JW, Kristjansson B, Saltzman MJ (eds) Operations research and cyber-infrastructure. Springer, Berlin, pp 195–217Google Scholar
  29. Engau A, Anjos MF, Vannelli A (2010a) An improved interior-point cutting-plane method for binary quadratic optimization. Electron Notes Discret Math 36:743–750Google Scholar
  30. Engau A, Anjos MF, Vannelli A (2010b) On interior-point warmstarts for linear and combinatorial optimization. SIAM J Optim 10(4):1828–1861Google Scholar
  31. Engau A, Anjos MF, Vannelli A (2012) On handling cutting-planes in interior-point methods for solving semidefinite relaxations of binary quadratic optimization problems. Optim Methods Softw 27(3): 539–559Google Scholar
  32. Fischer I, Gruber G, Rendl F, Sotirov R (2006) Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and equipartition. Math Program 105(2–3, Ser. B):451–469Google Scholar
  33. Fortunato S (2010) Community detection in graphs. Phys Rep 486(3–5):75–174MathSciNetCrossRefGoogle Scholar
  34. Gondzio J, Grothey A (2008) A new unblocking technique to warmstart interior point methods based on sensitivity analysis. SIAM J Optim 19(3):1184–1210MathSciNetMATHCrossRefGoogle Scholar
  35. Gruber G, Rendl F (2003) Computational experience with stable set relaxations. SIAM J Optim 13(4): 1014–1028Google Scholar
  36. Gvozdenović N, Laurent M (2008a) Computing semidefinite programming lower bounds for the (fractional) chromatic number via block-diagonalization. SIAM J Optim 19(2):592–615Google Scholar
  37. Gvozdenović N, Laurent M (2008b) The operator \(\Psi \) for the chromatic number of a graph. SIAM J Optim 19(2):572–591Google Scholar
  38. Hamzaoglu I, Patel JH (1998) Test set compaction algorithms for combinational circuits. In: Proceedings of the 1998 IEEE/ACM international conference on computer-aided design (ICCAD ’98), New York, NY, USA, pp 283–289Google Scholar
  39. Helmberg C (2004) A cutting plane algorithm for large scale semidefinite relaxations. In: The sharpest cut. SIAM, Philadelphia, PA, pp 233–256Google Scholar
  40. Helmberg C, Rendl F (1998) Solving quadratic \((0,1)\)-problems by semidefinite programs and cutting planes. Math Program 82(3, Ser. A):291–315Google Scholar
  41. John E, Yıldırım EA (2008) Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension. Comput Optim Appl 41(2):151–183MathSciNetMATHCrossRefGoogle Scholar
  42. Johnson DS, Trick MA, eds (1996) Cliques, coloring, and satisfiability. DIMACS series in discrete mathematics and theoretical computer science, 26. American Mathematical Society, Providence, RIGoogle Scholar
  43. Jung J, O’Leary D, Tits A (2012) Adaptive constraint reduction for convex quadratic programming. Comput Optim Appl 51:125–157MathSciNetMATHCrossRefGoogle Scholar
  44. Kaliski JA, Ye Y (1993) A short-cut potential reduction algorithm for linear programming. Manag Sci 39(6):757–776MATHCrossRefGoogle Scholar
  45. Karp RM (1972) Reducibility among combinatorial problems. In Complexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972). Plenum, New York, pp 85–103Google Scholar
  46. Laurent M, Rendl F (2005) Integer programming and semidefinite programming. In: Aardal K, Nemhauser GL, Weismantel R (eds) Discrete optimization. Handbooks in operations research and management science, vol 12. Elsevier, Amsterdam, pp 393–514Google Scholar
  47. Lovász L, Schrijver A (1991) Cones of matrices and set-functions and 0–1 optimization. SIAM J Optim 1(2):166–190MathSciNetMATHCrossRefGoogle Scholar
  48. McEliece RJ, Rodemich ER, Rumsey HC Jr (1978) The Lovász bound and some generalizations. J Combin Inform Syst Sci 3(3):134–152MathSciNetMATHGoogle Scholar
  49. Mitchell JE (2000) Computational experience with an interior point cutting plane algorithm. SIAM J Optim 10(4):1212–1227MathSciNetMATHCrossRefGoogle Scholar
  50. Mitchell JE (2009) Cutting plane methods and subgradient methods. In: Oskoorouchi M (ed) Tutorials in operations research Chapter 2, INFORMS, pp 34–61Google Scholar
  51. Mitchell JE, Borchers B (1996) Solving real-world linear ordering problems using a primal-dual interior point cutting plane method. Ann Oper Res 62:253–276MathSciNetMATHCrossRefGoogle Scholar
  52. Nesterov Y, Nemirovskii A (1994) Interior-point polynomial algorithms in convex programming volume 13 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PAGoogle Scholar
  53. Palagi L, Piccialli V, Rendl F, Rinaldi G, Wiegele A (2011) Computational approaches to max-cut. In: Handbook of semidefinite, conic and polynomial optimization: theory, algorithms, software and applications. International Series in Operations Research and Management Science. Springer, New York, pp 821–847Google Scholar
  54. Peña J, Vera J, Zuluaga LF (2007) Computing the stability number of a graph via linear and semidefinite programming. SIAM J Optim 18(1):87–105MathSciNetMATHCrossRefGoogle Scholar
  55. Rendl F (2012) Matrix relaxations in combinatorial optimization. In: Mixed integer nonlinear programming, vol 154. The IMA volumes in mathematics and its applications. Springer, Berlin, pp 483–511Google Scholar
  56. Rendl F, Rinaldi G, Wiegele A (2010) Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math Program 121(2, Ser. A):307–355Google Scholar
  57. Rhodes N, Willett P, Calvet A, Dunbar JB, Humblet C (2003) Clip: similarity searching of 3D databases using clique detection. J Chem Inf Comput Sci 43(2):443–448CrossRefGoogle Scholar
  58. Schrijver A (1979) A comparison of the Delsarte and Lovász bounds. IEEE Trans Inf Theory 25(4):425–429MathSciNetMATHCrossRefGoogle Scholar
  59. Sherali HD, Adams WP (1990) A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J Discret Math 3(3):411–430MathSciNetMATHCrossRefGoogle Scholar
  60. Sloane NJA (1989) Unsolved problems in graph theory arising from the study of codes. Graph Theory Notes NY 18:11–20Google Scholar
  61. Szegedy M, (1994) A note on the Theta number of Lovász and the generalized Delsarte bound. In: 35th Annual symposium on foundations of computer science, 20–22 November 1994, Santa Fe. New Mexico, USA, pp 36–39Google Scholar
  62. Tanay A, Sharan R, Shamir R (2002) Discovering statistically significant biclusters in gene expression data. Bioinformatics 1:S136–S144CrossRefGoogle Scholar
  63. Tütüncü RH, Toh K, Todd MJ (2003) Solving semidefinite-quadratic-linear programs using SDPT3. Math Program 95(2, Ser. B):189–217Google Scholar
  64. Vandenberghe L, Boyd S (1996) Semidefinite programming. SIAM Rev 38(1):49–95MathSciNetMATHCrossRefGoogle Scholar
  65. Wen Z, Goldfarb D, Yin W (2010) Alternating direction augmented Lagrangian methods for semidefinite programming. Math Program Comput 2(3–4):203–230MathSciNetMATHCrossRefGoogle Scholar
  66. Zhao XY, Sun D, Toh KC (2010) A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J Optim 20(4):1737–1765MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander Engau
    • 1
  • Miguel F. Anjos
    • 2
  • Immanuel Bomze
    • 3
  1. 1.University of Colorado DenverDenverUSA
  2. 2.GERAD & École Polytechnique de MontréalMontréalCanada
  3. 3.University of ViennaViennaAustria

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