Skip to main content
SpringerLink
Log in

On Mixed Integer Nonsmooth Optimization

  • Chapter
  • First Online:
Numerical Nonsmooth Optimization

Abstract

In this chapter we review some deterministic solution methods for convex mixed integer nonsmooth optimization problems. The methods are branch and bound, outer approximation, extended cutting plane, extended supporting hyperplane and extended level bundle method. Nonsmoothness is taken into account by using Clarke subgradients as a substitute for the classical gradient. Ideas for convergence proofs are given as well as references where the details can be found. We also consider how some algorithms can be modified in order to solve nonconvex problems including f ∘-pseudoconvex functions or even f ∘-quasiconvex constraints.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bagirov, A., Karmitsa, N., Mäkelä, M.M.: Introduction to Nonsmooth Optimization: Theory, Practice and Software. Springer, Berlin (2014)

    Google Scholar 

  2. Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biegler, L.T., Grossmann, I.E., Westerberg, A.W.: Systematic Methods for Chemical Process Design. Prentice Hall, Upper Saddle River (1997)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Chaib, A.E., Bouchekara, H.R.E.H., Mehasni, R., Abido, M.A.: Optimal power flow with emission and non-smooth cost functions using backtracking search optimization algorithm. Electr. Power Energy Syst. 81, 64–77 (2016)

    Article  Google Scholar 

  7. Clarke, F. H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)

    MATH  Google Scholar 

  8. CPLEX. https://www.ibm.com/analytics/cplex-optimizer

  9. Crouzeix, J.-P., Lindberg, P. O.: Additively decomposed quasiconvex functions. Math. Program. 35, 42–57 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dakin, R.J.: A tree-search algorithm for mixed integer programming problems. Comput. J. 8, 250–255 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  11. Delfino, A., de Oliveira, W.: Outer-approximation algorithms for nonsmooth convex MINLP problems. Optimization 67(6), 797–819 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. de Oliveira, W.: Regularized optimization methods for convex MINLP problems. TOP 24, 665–692 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. de Oliveira, W., Sagastizábal, C.: Bundle methods in the XXIst century: a bird’s eye view. Pesquisa Operacional 34, 647–670 (2014)

    Article  Google Scholar 

  14. de Oliveira, W., Solodov, M.: A doubly stabilized bundle method for nonsmooth convex optimization. Math. Program. 156(1), 125–159 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Drewes, S., Ulbrich, S.: Subgradient based outer approximation for mixed integer second order cone programming. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, pp. 41–59. Springer, New York (2012)

    Chapter  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Emet, S., Westerlund, T.: Comparisons of solving a chromatographic separation problem using MINLP methods. Comput. Chem. Eng. 28, 673–682 (2004)

    Article  Google Scholar 

  18. Eronen, V.-P., Mäkelä, M.M., Westerlund, T.: On the generalization of ECP and OA methods to nonsmooth MINLP problems. Optimization 63(7), 1057–1073 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Eronen, V.-P., Mäkelä, M.M., Westerlund, T.: Extended cutting-plane method for a class of nonsmooth nonconvex MINLP problems. Optimization 64(3), 641–661 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Eronen, V.-P., Kronqvist, J., Westerlund, T., Mäkelä, M.M., Karmitsa, N.: Method for solving generalized convex nonsmooth mixed-integer nonlinear programming problems. J. Glob. Optim. 69(2), 443–459 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fernandes, F.P., Costa, M.F.P., Fernandes, E.M.G.P.: Branch and bound based coordinate search filter algorithm for nonsmooth nonconvex mixed-integer nonlinear programming problems. In: Murgante, B., et al. (eds.) Computational Science and Its Applications—ICCSA 2014. ICCSA 2014. Lecture Notes in Computer Science, vol. 8580, pp. 140–153. Springer, Cham (2014)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. GAMS. https://www.gams.com/

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Gurobi. http://www.gurobi.com/

  26. Hamzeei, M., Luedtke, J.: Linearization-based algorithms for mixed-integer nonlinear programs with convex continuous relaxation. J. Glob. Optim. 59, 343–365 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kelley, J.E.: The cutting-plane method for solving convex programs. J. SIAM 8, 703–712 (1960)

    MathSciNet  MATH  Google Scholar 

  28. Khoury, G.A., Smadbeck, J., Kieslich, C.A., Floudas, C.A.: Protein folding and de novo protein design for biotechnological applications. Trends Biotechnol. 32(2), 99–109 (2014)

    Article  Google Scholar 

  29. Kravanja, S., Šilih, S., Kravanja, Z.: The multilevel MINLP optimization approach to structural synthesis: the simultaneous topology, material, standard and rounded dimension optimization. Adv. Eng. Softw. 36(9), 568–583 (2005)

    Article  MATH  Google Scholar 

  30. Kronqvist, J., Lundell, A., Westerlund, T.: The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. J. Glob. Optim. 64, 249–272 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Land, A.H., Doig, A.G.: An automatic method of solving discrete programming problems. Econometrica 28(3), 497–520 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lemaréchal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Math. Program. 69, 111–147 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liuzzi, G., Lucidi, S., Rinaldi, F.: Derivative-free methods for mixed-integer constrained optimization problems. J. Optim. Theory Appl. 164(3), 933–965 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mäkelä, M.M.: Survey of bundle methods for nonsmooth optimization. Optim. Methods Softw. 17(1), 1–29 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. MINLP Library. http://www.minlplib.org/instances.html

  36. Müller, J., Shoemaker, C.A., Piché, R.: SO-MI: a surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems. Comput. Oper. Res. 40, 1383–1400 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Newby, E., Ali, M.M.: A trust-region-based derivative free algorithm for mixed integer programming. Comput. Optim. Appl. 60, 199–229 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pörn, R., Westerlund, T.: A cutting-plane method for minimizing pseudo-convex functions in the mixed integer case. Comput. Chem. Eng. 24, 2655–2665 (2000)

    Article  Google Scholar 

  39. Pörn, R., Nissfolk, O., Jansson, F., Westerlund, T.: The Coulomb glass—modeling and computational experience with a large scale 0–1 QP problem. Comput. Aided Chem. Eng. 29, 658–662 (2011)

    Article  Google Scholar 

  40. Quesada, I., Grossmann, I.E.: An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1999)

    Article  Google Scholar 

  41. Rios, L.M., Sahinidis, N.V.: Derivative-free optimization: a review of algorithms and comparison of software implementations. J. Glob. Optim. 56(3), 1247–1293 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Schmidt, M., Steinbach, M.C., Willert, B.M.: A primal heuristic for nonsmooth mixed integer nonlinear optimization. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization, pp. 295–320. Springer, Berlin (2013)

    Chapter  MATH  Google Scholar 

  43. Shor, N.Z.: Minimization Methods for Non-Differentiable Functions. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  44. van Ackooij, W., de Oliveira, W.: Level bundle methods for constrained convex optimization with various oracles. Comput. Optim. Appl. 57, 555–597 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Veinott Jr, A.F.: The supporting hyperplane method for unimodal programming. Oper. Res. 15(1), 147–152 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wei, Z., Ali, M.M.: Convex mixed integer nonlinear programming problems and an outer approximation algorithm. J. Glob. Optim. 63(2), 213–227 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Wei, Z., Ali, M.M.: Outer approximation algorithm for one class of convex mixed-integer nonlinear programming problems with partial differentiability. J. Optim. Theory Appl. 167(2), 644–652 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  48. Westerlund, T., Pettersson, F.: An extended cutting-plane method for solving convex MINLP problems. Comput. Chem. Eng. 19(Supplement 1), 131–136 (1995)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  51. Westerlund, T., Eronen, V.-P., Mäkelä, M.M.: On solving generalized convex MINLP problems using supporting hyperplane techniques. J. Glob. Optim. 71(4), 987–1011 (2018). https://doi.org/10.1007/s10898-018-0644-z

    Article  MathSciNet  MATH  Google Scholar 

  52. Zioutas, G., Chatzinakos, C., Nguyen, T.D., Pitsoulis, L.: Optimization techniques for multivariate least trimmed absolute deviation estimation. J. Comb. Optim. 34, 781–797 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Turku, Turku, Finland

    Ville-Pekka Eronen, Tapio Westerlund & Marko M. Mäkelä

Authors
  1. Ville-Pekka Eronen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tapio Westerlund
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marko M. Mäkelä
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ville-Pekka Eronen .

Editor information

Editors and Affiliations

  1. School of Science, Engineering and Information Technology, Federation University Australia, Ballarat, VIC, Australia

    Adil M. Bagirov

  2. Department of Informatics, Modeling, Electronics and System Engineering, University of Calabria, Rende (CS), Italy

    Manlio Gaudioso

  3. Department of Mathematics and Statistics, University of Turku, Turku, Finland

    Napsu Karmitsa

  4. Department of Mathematics and Statistics, University of Turku, Turku, Finland

    Marko M. Mäkelä

  5. School of Science, Engineering and Information Technology, Federation University Australia, Ballarat, VIC, Australia

    Sona Taheri

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Eronen, VP., Westerlund, T., Mäkelä, M.M. (2020). On Mixed Integer Nonsmooth Optimization. In: Bagirov, A., Gaudioso, M., Karmitsa, N., Mäkelä, M., Taheri, S. (eds) Numerical Nonsmooth Optimization. Springer, Cham. https://doi.org/10.1007/978-3-030-34910-3_16

Download citation

Publish with us

Policies and ethics

Search

Navigation