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SpeeDP: an algorithm to compute SDP bounds for very large Max-Cut instances

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Abstract

We consider low-rank semidefinite programming (LRSDP) relaxations of unconstrained \(\{-1,1\}\) quadratic problems (or, equivalently, of Max-Cut problems) that can be formulated as the non-convex nonlinear programming problem of minimizing a quadratic function subject to separable quadratic equality constraints. We prove the equivalence of the LRSDP problem with the unconstrained minimization of a new merit function and we define an efficient and globally convergent algorithm, called SpeeDP, for finding critical points of the LRSDP problem. We provide evidence of the effectiveness of SpeeDP by comparing it with other existing codes on an extended set of instances of the Max-Cut problem. We further include SpeeDP within a simply modified version of the Goemans–Williamson algorithm and we show that the corresponding heuristic SpeeDP-MC can generate high-quality cuts for very large, sparse graphs of size up to a million nodes in a few hours.

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Notes

  1. Actually, simple bounds like, e.g., the sum of all positive edge weights, can always be easily provided, but they are evidently useless to evaluate the quality of a heuristic solution.

  2. http://www-unix.mcs.anl.gov/DSDP/.

  3. http://www-user.tu-chemnitz.de/~helmberg/.

  4. http://dollar.biz.uiowa.edu/~sburer/software/SDPLR.

  5. http://euler.nmt.edu/~brian/sdplib.

References

  1. Barvinok, A.: Problems of distance geometry and convex properties of quadratic maps. Discret. Comput. Geom. 13, 189–202 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benson, H.Y., Vanderbei, R.J.: On formulating semidefinite programming problems as smooth convex nonlinear optimization problems. Technical Report ORFE 1999–01, Princeton University, NJ (1999)

  3. Benson, S.J., Ye, Y., Zhang, X.: Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10(2), 443–461 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  5. Burer, S., Monteiro, R.D., Zhang, Y.: Solving a class of semidefinite programs via nonlinear programming. Math. Program. 93, 97–122 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burer, S., Monteiro, R.D.C.: A projected gradient algorithm for solving the maxcut sdp relaxation. Optim. Methods Softw. 15(3–4), 175–200 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burer, S., Monteiro, R.D.C.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. 95, 329–357 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Delorme, C., Poljak, S.: Laplacian eigenvalues and the maximum cut problem. Math. Program. 62(3), 557–574 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dolan, E., Morè, J.: Benchmarking optimization software with performance profile. Math. Program. 91, 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fischer, I., Gruber, G., Rendl, F., Sotirov, R.: Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and equipartition. Math. Program. 105(2–3), 451–469 (2006)

    Google Scholar 

  11. Fletcher, R.: Semi-definite matrix constraints in optimization. SIAM J. Cont. Optim. 23, 493–513 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grippo, L., Palagi, L., Piacentini, M., Piccialli, V.: An unconstrained approach for solving low rank SDP relaxations of \(\{-1,1\}\) quadratic problems. Technical Report 1.13, Dip. di Informatica e Sistemistica A. Ruberti, Sapienza Università di Roma (2009)

  14. Grippo, L., Palagi, L., Piacentini, M., Piccialli, V., Rinaldi, G.: SpeeDP: a new algorithm to compute the SDP relaxations of Max-Cut for very large graphs. Technical Report DII-UTOVRM Technical Report 13.10, University of Rome Tor Vergata (2010)

  15. Grippo, L., Palagi, L., Piccialli, V.: Necessary and sufficient global optimality conditions for NLP reformulations of linear SDP problems. J. Glob. Optim. 44(3), 339–348 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grippo, L., Palagi, L., Piccialli, V.: An unconstrained minimization method for solving low-rank SDP relaxations of the maxcut problem. Math. Program. 126, 119–146 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grippo, L., Sciandrone, M.: Nonmonotone globalization techniques for the Barzilai–Borwein gradient method. Comput. Optim. Appl. 23, 143–169 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grone, R., Pierce, S., Watkins, W.: Extremal correlation matrices. Linear Algebra Appl. 134, 63–70 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10, 673–696 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Homer, S., Peinado, M.: Design and performance of parallel and distributed approximation algorithm for the Maxcut. J. Parallel Distrib. Comput. 46, 48–61 (1997)

    Article  Google Scholar 

  21. Journée, M., Bach, F., Absil, P., Sepulchre, R.: Low-rank optimization for semidefinite convex problems. SIAM J. Optim. 20(5), 2327–2351 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kim, S., Kojima, M.: Second order cone programming relaxation of nonconvex quadratic optimization problems. Optim. Methods Softw. 15, 201–224 (2000)

    Article  MathSciNet  Google Scholar 

  23. Laurent, M., Poljak, S., Rendl, F.: Connections between semidefinite relaxations of the Max-Cut and stable set problems. Math. Program. 77, 225–246 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Liers, F., Jünger, M., Reinelt, G., Rinaldi, G.: Computing exact ground states of hard Ising spin glass problems by branch-and-cut. In: Hartmann, A., Rieger, H. (eds.) New Optimization Algorithms in Physics, pp. 47–69. Wiley, London (2004)

    Google Scholar 

  25. Mittelmann, H.: An independent benchmarking of SDP and SOCP solvers. Math. Program. 95, 407–430 (2003)

    Google Scholar 

  26. Muramatsu, M., Suzuki, T.: A new second-order cone programming relaxation for max-cut problems. J. Oper. Res. Jpn 46, 2003 (2001)

  27. Nesterov, Y.: Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion Papers 1997019, Université Catholique de Louvain, Center for Operations Research and Econometrics (CORE) (1997)

  28. Nesterov, Y.: Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 9(1–3), 141–160 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pataki, G.: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23, 339–358 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Poljak, S., Rendl, F.: Solving the Max-Cut problem using eigenvalues. Discret. Appl. Math. 62(1–3), 249–278 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Poljak, S., Rendl, F., Wolkowicz, H.: A recipe for semidefinite relaxation for 0-1 quadratic programming. J. Glob. Optim. 7, 51–73 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rendl, F., Rinaldi, G., Wiegele, A.: Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121(2), 307–335 (2010)

    Google Scholar 

  33. Rinaldi, G.: Rudy: A graph generator. http://www-user.tu-chemnitz.de/~helmberg/sdp_software.html (1998)

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Acknowledgments

We would like to thank the three anonymous referees for their valuable remarks and suggestions that helped us to improve the writing of this paper.

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Authors and Affiliations

  1. Dipartimento di Ingegneria informatica automatica e gestionale A. Ruberti, Sapienza Università di Roma, via Ariosto, 25, 00185 , Rome, Italy

    Luigi Grippo, Laura Palagi & Mauro Piacentini

  2. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università degli Studi di Roma Tor Vergata, via del Politecnico, 1, 00133 , Rome, Italy

    Veronica Piccialli

  3. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” del CNR, viale Manzoni, 30, 00185 , Rome, Italy

    Giovanni Rinaldi

Authors
  1. Luigi Grippo
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  2. Laura Palagi
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  3. Mauro Piacentini
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  4. Veronica Piccialli
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  5. Giovanni Rinaldi
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Correspondence to Veronica Piccialli.

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Grippo, L., Palagi, L., Piacentini, M. et al. SpeeDP: an algorithm to compute SDP bounds for very large Max-Cut instances. Math. Program. 136, 353–373 (2012). https://doi.org/10.1007/s10107-012-0593-0

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  • DOI: https://doi.org/10.1007/s10107-012-0593-0

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