Journal of Global Optimization

, Volume 68, Issue 3, pp 501–535 | Cite as

A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes

  • Kaisa Joki
  • Adil M. Bagirov
  • Napsu Karmitsa
  • Marko M. Mäkelä


In this paper, we develop a version of the bundle method to solve unconstrained difference of convex (DC) programming problems. It is assumed that a DC representation of the objective function is available. Our main idea is to utilize subgradients of both the first and second components in the DC representation. This subgradient information is gathered from some neighborhood of the current iteration point and it is used to build separately an approximation for each component in the DC representation. By combining these approximations we obtain a new nonconvex cutting plane model of the original objective function, which takes into account explicitly both the convex and the concave behavior of the objective function. We design the proximal bundle method for DC programming based on this new approach and prove the convergence of the method to an \(\varepsilon \)-critical point. The algorithm is tested using some academic test problems and the preliminary numerical results have shown the good performance of the new bundle method. An interesting fact is that the new algorithm finds nearly always the global solution in our test problems.


Nonsmooth optimization Nonconvex optimization Proximal bundle methods DC functions Cutting plane model 

Mathematics Subject Classification

90C26 49J52 65K05 



We are thankful to the anonymous referees for their valuable comments. We would also like to acknowledge Professors M. Gaudioso and A. Fuduli for kindly providing us the codes of NCVX and NCVX-penalty methods. This work has been financially supported by the Jenny and Antti Wihuri Foundation, the Turku University Foundation, the University of Turku, the Academy of Finland (project number: 289500) and the Australian Research Council’s Discovery Projects funding scheme (project number: DP140103213).


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Kaisa Joki
    • 1
  • Adil M. Bagirov
    • 2
  • Napsu Karmitsa
    • 1
  • Marko M. Mäkelä
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  2. 2.Faculty of Science and TechnologyFederation University AustraliaBallaratAustralia

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