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Globally Convergent Cutting Plane Method for Nonconvex Nonsmooth Minimization

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Abstract

Nowadays, solving nonsmooth (not necessarily differentiable) optimization problems plays a very important role in many areas of industrial applications. Most of the algorithms developed so far deal only with nonsmooth convex functions. In this paper, we propose a new algorithm for solving nonsmooth optimization problems that are not assumed to be convex. The algorithm combines the traditional cutting plane method with some features of bundle methods, and the search direction calculation of feasible direction interior point algorithm (Herskovits, J. Optim. Theory Appl. 99(1):121–146, 1998). The algorithm to be presented generates a sequence of interior points to the epigraph of the objective function. The accumulation points of this sequence are solutions to the original problem. We prove the global convergence of the method for locally Lipschitz continuous functions and give some preliminary results from numerical experiments.

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Authors and Affiliations

  1. Department of Mathematics, University of Turku, 20014, Turku, Finland

    Napsu Karmitsa

  2. Mechanical Engineering Program, COPPE—Federal University of Rio de Janeiro, Caixa Postal 68503, 21945 970, Rio de Janeiro, Brazil

    Mario Tanaka Filho & José Herskovits

Authors
  1. Napsu Karmitsa
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  2. Mario Tanaka Filho
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  3. José Herskovits
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Corresponding author

Correspondence to Napsu Karmitsa.

Additional information

Communicated by P.M. Pardalos.

The authors would like to thank Prof. Marko M. Mäkelä (University of Turku, Finland) for the valuable comments and Prof. Tapio Westerlund (Abo Akademi University, Finland) for financial support. The work was financially supported by the Research Councils CAPES, CNPq and Faperj (Brazil), COPPE/Federal University of Rio de Janeiro (Brazil), University of Turku (Finland), Magnus Ehrnrooth foundation (Finland), and the Academy of Finland (Project No. 127992).

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Karmitsa, N., Tanaka Filho, M. & Herskovits, J. Globally Convergent Cutting Plane Method for Nonconvex Nonsmooth Minimization. J Optim Theory Appl 148, 528–549 (2011). https://doi.org/10.1007/s10957-010-9766-2

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  • DOI: https://doi.org/10.1007/s10957-010-9766-2

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