Skip to main content
SpringerLink
Log in

Inexact solution of NLP subproblems in MINLP

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In the context of convex mixed integer nonlinear programming (MINLP), we investigate how the outer approximation method and the generalized Benders decomposition method are affected when the respective nonlinear programming (NLP) subproblems are solved inexactly. We show that the cuts in the corresponding master problems can be changed to incorporate the inexact residuals, still rendering equivalence and finiteness in the limit case. Some numerical results will be presented to illustrate the behavior of the methods under NLP subproblem inexactness.

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

Similar content being viewed by others

References

  1. CMU/IBM MINLP Project. http://egon.cheme.cmu.edu/ibm/page.htm

  2. IBM ILOG CPLEX. http://www-01.ibm.com/software/integration/optimization/cplex-optimizer. Version 12.4, 2012

  3. Ahlatçıoğlu, A., Guignard, M.: Convex hull relaxation (CHR) for convex and nonconvex MINLP problems with linear constraints (2011)

  4. Belotti P., Lee J., Liberti L., Margot F., Wächter A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24, 597–634 (2009)

    Article  Google Scholar 

  5. Benders J.F.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4, 238–252 (1962)

    Article  Google Scholar 

  6. Bonami P., Biegler L.T., Conn A.R., Cornuéjols G., Grossmann I.E., Laird C.D., Lee J., Lodi A., Margot F., Sawaya N., Wächter A.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Optim. 5, 186–204 (2008)

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. Bonami P., Gonçalves J.P.M.: Heuristics for convex mixed integer nonlinear programs. Comput. Optim. Appl. 51, 729–747 (2012)

    Article  Google Scholar 

  9. Castillo I., Westerlund J., Emet S., Westerlund T.: Optimization of block layout design problems with unequal areas: A comparison of MILP and MINLP optimization methods. Comput. Chem. Eng. 30, 54–69 (2005)

    Article  Google Scholar 

  10. Duran M.A., Grossmann I.E.: An out-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)

    Article  Google Scholar 

  11. Fletcher R., Leyffer S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349 (1994)

    Article  Google Scholar 

  12. Flippo O.E., Rinnooy Kan A.H.G.: Decomposition in general mathematical programming. Math. Program. 60, 361–382 (1993)

    Article  Google Scholar 

  13. Geoffrion A.M.: Generalized Benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972)

    Article  Google Scholar 

  14. Grossmann I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)

    Article  Google Scholar 

  15. Kelley J.E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8, 703–712 (1960)

    Article  Google Scholar 

  16. Leyffer, S.: MacMINLP. http://wiki.mcs.anl.gov/leyffer/index.php/MacMINLP

  17. Nannicini, G., Belotti, P., Lee, J., Linderoth, J., Margot, F., Wächter, A.: A probing algorithm for MINLPs with early detection of failures by SVM. In: CPAIOR 2011: The 8th International Conference on Integration of Artificial Intelligence and Operations Research, volume 6697 of Lecture Notes in Computer Science, pp. 154–169. Springer, New York (2011)

  18. Quesada I., Grossmann I.E.: An LP/NLP based branch and bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1992)

    Article  Google Scholar 

  19. Tawarmalani M., Sahinidis N.V.: Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Math. Program. 99, 563–591 (2004)

    Article  Google Scholar 

  20. Westerlund T., Pettersson F.: An extended cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, 131–136 (1995)

    Article  Google Scholar 

  21. Westerlund T., Pörn R.: Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3, 253–280 (2002)

    Article  Google Scholar 

  22. Westerlund T., Skrifvars H., Harjunkoski I., Pörn R.: An extended cutting plane method for a class of non-convex MINLP problems. Comput. Chem. Eng. 22, 357–365 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Coimbra, 3001-454, Coimbra, Portugal

    M. Li

  2. CMUC, Department of Mathematics, University of Coimbra, 3001-454, Coimbra, Portugal

    L. N. Vicente

Authors
  1. M. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. N. Vicente
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. N. Vicente.

Additional information

M. Li was supported by FCT under the scholarship SFRH/BD/33369/2008.

L. N. Vicente was supported by FCT under the grant PTDC/MAT/098214/2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, M., Vicente, L.N. Inexact solution of NLP subproblems in MINLP. J Glob Optim 55, 877–899 (2013). https://doi.org/10.1007/s10898-012-0010-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-0010-5

Keywords

Search

Navigation