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

Author

Listed:
  • Ville-Pekka Eronen

    (University of Turku)

  • Jan Kronqvist

    (Åbo Akademi)

  • Tapio Westerlund

    (Åbo Akademi)

  • Marko M. Mäkelä

    (University of Turku)

  • Napsu Karmitsa

    (University of Turku)

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 $$f^{\circ }$$ f ∘ -pseudoconvex. With some additional assumptions, the constraint functions may be $$f^{\circ }$$ f ∘ -quasiconvex.

Suggested Citation

  • Ville-Pekka Eronen & Jan Kronqvist & Tapio Westerlund & Marko M. Mäkelä & Napsu Karmitsa, 2017. "Method for solving generalized convex nonsmooth mixed-integer nonlinear programming problems," Journal of Global Optimization, Springer, vol. 69(2), pages 443-459, October.
    • Handle: RePEc:spr:jglopt:v:69:y:2017:i:2:d:10.1007_s10898-017-0528-7
    DOI: 10.1007/s10898-017-0528-7
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    References listed on IDEAS

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    1. Arthur F. Veinott, 1967. "The Supporting Hyperplane Method for Unimodal Programming," Operations Research, INFORMS, vol. 15(1), pages 147-152, February.
    2. Adil Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2014. "Introduction to Nonsmooth Optimization," Springer Books, Springer, edition 127, number 978-3-319-08114-4, June.
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    Cited by:

    1. Tapio Westerlund & Ville-Pekka Eronen & Marko M. Mäkelä, 2018. "On solving generalized convex MINLP problems using supporting hyperplane techniques," Journal of Global Optimization, Springer, vol. 71(4), pages 987-1011, August.
    2. Felipe Serrano & Robert Schwarz & Ambros Gleixner, 2020. "On the relation between the extended supporting hyperplane algorithm and Kelley’s cutting plane algorithm," Journal of Global Optimization, Springer, vol. 78(1), pages 161-179, September.

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