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

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.

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

Fig. 1
Fig. 2
Fig. 3

References

  1. Alizadeh F (1995) Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J Optim 5(1):13–51

    MathSciNet  MATH  Article  Google 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):611–617

    MathSciNet  MATH  Article  Google 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):295–324

    Google Scholar 

  4. Barahona F, Epstein R, Weintraub A (1992) Habitat dispersion in forest planning and the stable set problem. Oper Res 40(1):S14–S21

    Article  Google 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):371–399

    MathSciNet  MATH  Article  Google Scholar 

  6. Bomze IM (2012) Copositive optimization-recent developments and applications. Eur J Oper Res 216(3):509–520

    MathSciNet  MATH  Article  Google Scholar 

  7. Bomze IM, De Klerk E (2002) Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J Global Optim 24(2):163–185

    MathSciNet  MATH  Article  Google 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):13–32

    Google 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):31–64

    Google 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):423–445

    MathSciNet  MATH  Article  Google Scholar 

  12. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge

    Google Scholar 

  13. Burer S (2010) Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math Program Comput 2(1):1–19

    MathSciNet  MATH  Article  Google 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 201–218

  15. Dantzig GB, Ye Y (1991) A build-up interior method for linear programming: affine scaling form. Technical report, Stanford University/The University of Iowa

  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, Dordrecht

  18. de Klerk E (2010) Exploiting special structure in semidefinite programming: a survey of theory and applications. Eur J Oper Res 201:1–10

    MATH  Article  Google Scholar 

  19. de Klerk E, Pasechnik DV (2002) Approximation of the stability number of a graph via copositive programming. SIAM J Optim 12(4):875–892

    MathSciNet  MATH  Article  Google 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 166–185

  21. Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91(2, Ser. A):201–213

    Google Scholar 

  22. Dukanovic I, Rendl F (2007) Semidefinite programming relaxations for graph coloring and maximal clique problems. Math Program 109(2–3, Ser. B):345–365

    Google Scholar 

  23. Dukanovic I, Rendl F (2010) Copositive programming motivated bounds on the stability and the chromatic numbers. Math Program 121(2, Ser. A):249–268

    Google 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 3–20

  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):45–72

    MathSciNet  MATH  Article  Google 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 471–498

  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, Canada

  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 195–217

  29. Engau A, Anjos MF, Vannelli A (2010a) An improved interior-point cutting-plane method for binary quadratic optimization. Electron Notes Discret Math 36:743–750

    Google Scholar 

  30. Engau A, Anjos MF, Vannelli A (2010b) On interior-point warmstarts for linear and combinatorial optimization. SIAM J Optim 10(4):1828–1861

    Google 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): 539–559

    Google 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):451–469

    Google Scholar 

  33. Fortunato S (2010) Community detection in graphs. Phys Rep 486(3–5):75–174

    MathSciNet  Article  Google 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):1184–1210

    MathSciNet  MATH  Article  Google Scholar 

  35. Gruber G, Rendl F (2003) Computational experience with stable set relaxations. SIAM J Optim 13(4): 1014–1028

    Google 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):592–615

    Google Scholar 

  37. Gvozdenović N, Laurent M (2008b) The operator \(\Psi \) for the chromatic number of a graph. SIAM J Optim 19(2):572–591

    Google 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 283–289

  39. Helmberg C (2004) A cutting plane algorithm for large scale semidefinite relaxations. In: The sharpest cut. SIAM, Philadelphia, PA, pp 233–256

  40. Helmberg C, Rendl F (1998) Solving quadratic \((0,1)\)-problems by semidefinite programs and cutting planes. Math Program 82(3, Ser. A):291–315

    Google 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):151–183

    MathSciNet  MATH  Article  Google 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, RI

  43. Jung J, O’Leary D, Tits A (2012) Adaptive constraint reduction for convex quadratic programming. Comput Optim Appl 51:125–157

    MathSciNet  MATH  Article  Google Scholar 

  44. Kaliski JA, Ye Y (1993) A short-cut potential reduction algorithm for linear programming. Manag Sci 39(6):757–776

    MATH  Article  Google 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 85–103

  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 393–514

  47. Lovász L, Schrijver A (1991) Cones of matrices and set-functions and 0–1 optimization. SIAM J Optim 1(2):166–190

    MathSciNet  MATH  Article  Google Scholar 

  48. McEliece RJ, Rodemich ER, Rumsey HC Jr (1978) The Lovász bound and some generalizations. J Combin Inform Syst Sci 3(3):134–152

    MathSciNet  MATH  Google Scholar 

  49. Mitchell JE (2000) Computational experience with an interior point cutting plane algorithm. SIAM J Optim 10(4):1212–1227

    MathSciNet  MATH  Article  Google Scholar 

  50. Mitchell JE (2009) Cutting plane methods and subgradient methods. In: Oskoorouchi M (ed) Tutorials in operations research Chapter 2, INFORMS, pp 34–61

  51. Mitchell JE, Borchers B (1996) Solving real-world linear ordering problems using a primal-dual interior point cutting plane method. Ann Oper Res 62:253–276

    MathSciNet  MATH  Article  Google 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, PA

  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 821–847

  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):87–105

    MathSciNet  MATH  Article  Google 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 483–511

  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):307–355

    Google 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):443–448

    Article  Google Scholar 

  58. Schrijver A (1979) A comparison of the Delsarte and Lovász bounds. IEEE Trans Inf Theory 25(4):425–429

    MathSciNet  MATH  Article  Google 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):411–430

    MathSciNet  MATH  Article  Google Scholar 

  60. Sloane NJA (1989) Unsolved problems in graph theory arising from the study of codes. Graph Theory Notes NY 18:11–20

    Google 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 36–39

  62. Tanay A, Sharan R, Shamir R (2002) Discovering statistically significant biclusters in gene expression data. Bioinformatics 1:S136–S144

    Article  Google Scholar 

  63. Tütüncü RH, Toh K, Todd MJ (2003) Solving semidefinite-quadratic-linear programs using SDPT3. Math Program 95(2, Ser. B):189–217

    Google Scholar 

  64. Vandenberghe L, Boyd S (1996) Semidefinite programming. SIAM Rev 38(1):49–95

    MathSciNet  MATH  Article  Google Scholar 

  65. Wen Z, Goldfarb D, Yin W (2010) Alternating direction augmented Lagrangian methods for semidefinite programming. Math Program Comput 2(3–4):203–230

    MathSciNet  MATH  Article  Google Scholar 

  66. Zhao XY, Sun D, Toh KC (2010) A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J Optim 20(4):1737–1765

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgments

We gratefully acknowledge the helpful comments by the associate editor and two anonymous referees that have allowed us to improve this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alexander Engau.

Additional information

Alexander Engau: Research partially supported by the DFG Emmy Noether project “Combinatorial Optimization in Physics (COPhy)” at the University of Cologne, Germany and by MITACS, a Network of Centres of Excellence for the Mathematical Sciences in Canada.

Miguel F. Anjos: Research partially supported by the Natural Sciences and Engineering Research Council of Canada, and by a Humboldt Research Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Engau, A., Anjos, M.F. & Bomze, I. Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem. Math Meth Oper Res 78, 35–59 (2013). https://doi.org/10.1007/s00186-013-0431-z

Download citation

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