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Journal of Global Optimization

, Volume 69, Issue 2, pp 443–459 | Cite as

Method for solving generalized convex nonsmooth mixed-integer nonlinear programming problems

  • Ville-Pekka EronenEmail author
  • Jan Kronqvist
  • Tapio Westerlund
  • Marko M. Mäkelä
  • Napsu Karmitsa
Article

Abstract

In this paper, we generalize the extended supporting hyperplane algorithm for a convex continuously differentiable mixed-integer nonlinear programming problem to solve a wider class of nonsmooth problems. The generalization is made by using the subgradients of the Clarke subdifferential instead of gradients. Consequently, all the functions in the problems are assumed to be locally Lipschitz continuous. The algorithm is shown to converge to a global minimum of an MINLP problem if the objective function is convex and the constraint functions are \(f^{\circ }\)-pseudoconvex. With some additional assumptions, the constraint functions may be \(f^{\circ }\)-quasiconvex.

Keywords

MINLP Extended supporting hyperplane method Convex optimization Nonsmooth optimization Clarke subdifferential Generalized convexities 

Mathematics Subject Classification

90C11 90C25 

Notes

Acknowledgements

This research was supported by the Grant No. 289500 and 294002 of the Academy of Finland. Jan Kronqvist would like to thank the Graduate School in Chemical Engineering.

References

  1. 1.
    Arnold, T., Henrion, R., Moller, A., Vigerske, S.: A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints. Pac. J. Optim. 10, 5–20 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bagirov, A., Mäkelä, M.M., Karmitsa, N.: Introduction to Nonsmooth Optimization: Theory, Practice and Software. Springer, Cham (2014)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bonami, P., Cournuéjols, G., Lodi, A., Margot, F.: A feasibility pump for mixed integer nonlinear programs. Math. Program. 119, 331–352 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)zbMATHGoogle Scholar
  6. 6.
    de Oliveira, W.: Regularized optimization methods for convex MINLP problems. TOP 24, 665–692 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Emet, S., Westerlund, T.: Comparisons of solving a chromatographic separation problem using MINLP methods. Comput. Chem. Eng. 28, 673–682 (2004)CrossRefGoogle Scholar
  8. 8.
    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)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kronqvist, J., Lundell, A., Westerlund, T.: The ESH algorithm for convex mixed-integer nonlinear programming. J. Glob. Optim. 64(2), 249–272 (2016)Google Scholar
  10. 10.
    Meller, R.D., Narayanan, V., Vance, P.H.: Optimal facility layout design. Oper. Res. Lett. 23, 117–127 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pörn, R.: Mixed-Integer Non-Linear Programming: Convexification Techniques and Algorithm Development. Ph.D. Thesis, Åbo Akademi University (2000)Google Scholar
  12. 12.
    Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2(1), 121–152 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Veinott Jr., A.F.: The supporting hyperplane method for unimodal programming. Oper. Res. 15(1), 147–152 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Westerlund, T., Pörn, R.: Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3, 253–280 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  2. 2.Optimization and Systems EngineeringÅbo AkademiTurkuFinland

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