Journal of Global Optimization

, Volume 59, Issue 4, pp 787–809 | Cite as

Level bundle-like algorithms for convex optimization

  • J. Y. Bello Cruz
  • W. de Oliveira


We propose two restricted memory level bundle-like algorithms for minimizing a convex function over a convex set. If the memory is restricted to one linearization of the objective function, then both algorithms are variations of the projected subgradient method. The first algorithm, proposed in Hilbert space, is a conceptual one. It is shown to be strongly convergent to the solution that lies closest to the initial iterate. Furthermore, the entire sequence of iterates generated by the algorithm is contained in a ball with diameter equal to the distance between the initial point and the solution set. The second algorithm is an implementable version. It mimics as much as possible the conceptual one in order to resemble convergence properties. The implementable algorithm is validated by numerical results on several two-stage stochastic linear programs.


Convex minimization Nonsmooth optimization Level bundle method Strong convergence 



The authors are grateful to the reviewers for their insightful comments and remarks. Research for this paper was partially supported by PROCAD-nf—UFG/UnB/IMPA research and PRONEX—CNPq-FAPERJ—Optimization research, and by project CAPES-MES-CUBA 226/2012 “Modelos de Otimização e Aplicações ”. The first author thanks CNPq and the Institute for Pure and Applied Mathematics (IMPA), in Rio de Janeiro, Brazil, where he was a visitor while working on this paper.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade Federal de GoiásGoiâniaBrazil
  2. 2.Instituto Nacional de Matemática Pura e AplicadaRio de JaneiroBrazil

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