Skip to main content
SpringerLink
Log in

Proximity search for 0-1 mixed-integer convex programming

  • Published:
Journal of Heuristics Aims and scope Submit manuscript

Abstract

In this paper we investigate the effects of replacing the objective function of a 0-1 mixed-integer convex program (MIP) with a “proximity” one, with the aim of using a black-box solver as a refinement heuristic. Our starting observation is that enumerative MIP methods naturally tend to explore a neighborhood around the solution of a relaxation. A better heuristic performance can however be expected by searching a neighborhood of an integer solution—a result that we obtain by just modifying the objective function of the problem at hand. The relationship of this approach with primal integer methods is also addressed. Promising computational results on different proof-of-concept implementations are presented, suggesting that proximity search can be quite effective in quickly refining a given feasible solution. This is particularly true when a sequence of similar MIPs has to be solved as, e.g., in a column-generation setting.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Achterberg, T., Berthold, T.: Improving the feasibility pump. Discret. Optim. 4(1), 77–86 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Achterberg, T., Berthold, T., Hendel, G.: Rounding and propagation heuristics for mixed integer programming. Op. Res. Proc. 2011, 71–76 (2012)

    Article  Google Scholar 

  • Bertacco, L., Fischetti, M., Lodi, A.: A feasibility pump heuristic for general mixed-integer problems. Discret. Optim. 4(1), 63–76 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Berthold, T.: Measuring the impact of primal heuristics. Oper. Res. Lett. 41(6), 611–614 (2013)

  • Boland, N.L., Eberhard, A.C., Engineer, F.G., Tsoukalas, A.: A new approach to the feasibility pump in mixed integer programming. SIAM J. Optim. 22(3), 831–861 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Bonami, P., Cornuéjols, G., Lodi, A., Margot, F.: A feasibility pump for mixed integer nonlinear programs. Math. Progr. 119(2), 331–352 (2009)

    Article  MATH  Google Scholar 

  • Brooks, J.P.: Support vector machines with the ramp loss and the hard margin loss. Op. Res. 59(2), 467–479 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • D’Ambrosio, C., Frangioni, A., Liberti, L., Lodi, A.: A storm of feasibility pumps for nonconvex MINLP. Math. Progr. 136(2), 375–402 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Daniilidis, A., Lemarechal, C.: On a primal-proximal heuristic in discrete optimization. Math. Progr. 104(1), 105–128 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Danna, E., Rothberg, E., Le Pape, C.: Exploring relaxation induced neighborhoods to improve MIP solutions. Math. Progr. 102(1), 71–90 (2005)

    Article  MATH  Google Scholar 

  • Fischetti, M., Glover, F., Lodi, A.: The feasibility pump. Math. Progr. 104(1), 91–104 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Fischetti, M., Lodi, A.: Local branching. Math. Progr. 98(1–3), 23–47 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Fischetti, M., Lodi, A.: Heuristics in Mixed Integer Programming, vol. 8, pp. 738–747. Wiley, New York (2011)

    Google Scholar 

  • Fischetti, M., Sartor, G., Zanette, A.: A MIP-and-refine matheuristic for smart grid energy management. Int. Trans. Op. Res., 1–11 (2013). doi:10.1111/itor.12034

  • Fischetti, M., Monaci, M.: Proximity search heuristics for wind farm optimal layout. Technical Report DEI, University of Padova (2014, submitted).

  • Glover, F.: Parametric branch and bound. OMEGA 6(2), 145–152 (1978)

    Article  Google Scholar 

  • Glover, F.: Parametric tabu search for mixed integer programs. Comput. Op. Res. 33(9), 2449–2494 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, Volume II: Advanced Theory and Bundle Methods, Volume 306 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1993)

  • Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010 mixed integer programming library version 5. Math. Progr. Comput. 3(2), 103–163 (2011)

  • Nocedal, J., Wright, S.: Numerical Optimization. Springer, Berlin (2009)

    Google Scholar 

  • Orlowski, S., Wessäly, R., Pióro, M., Tomaszewski, A.: SNDlib 1.0—survivable network design library. Networks 55(3), 276–286 (2010)

    Google Scholar 

  • Raack, C., Koster, A., Orlowski, S., Wessäly, R.: On cut-based inequalities for capacitated network design polyhedra. Networks 57(2), 141–156 (2011)

    MATH  MathSciNet  Google Scholar 

  • Rothberg, E.: An evolutionary algorithm for polishing mixed integer programming solutions. INFORMS J. Comput. 19(4), 534–541 (2007)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This research was supported by the Progetto di Ateneo on “Exploiting randomness in Mixed-Integer Linear Programming” of the University of Padova, and by MiUR, Italy (PRIN project “Mixed-Integer Nonlinear Optimization: Approaches and Applications”). We thank J.P. Brooks who provided us with the classification instances.

Author information

Authors and Affiliations

  1. DEI, University of Padova, Via Gradenigo 6/A, 35100, Padua, Italy

    Matteo Fischetti & Michele Monaci

Authors
  1. Matteo Fischetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michele Monaci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michele Monaci.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fischetti, M., Monaci, M. Proximity search for 0-1 mixed-integer convex programming. J Heuristics 20, 709–731 (2014). https://doi.org/10.1007/s10732-014-9266-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-014-9266-x

Keywords

Search

Navigation