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


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.


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

Mathematics Subject Classification

90C11 90C25 



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.


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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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