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Level bundle-like algorithms for convex optimization

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  • J. Bello Cruz
  • W. Oliveira

Abstract

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. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • J. Bello Cruz & W. Oliveira, 2014. "Level bundle-like algorithms for convex optimization," Journal of Global Optimization, Springer, vol. 59(4), pages 787-809, August.
    • Handle: RePEc:spr:jglopt:v:59:y:2014:i:4:p:787-809
    DOI: 10.1007/s10898-013-0096-4
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    References listed on IDEAS

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    1. Gerd Infanger (ed.), 2011. "Stochastic Programming," International Series in Operations Research and Management Science, Springer, number 978-1-4419-1642-6, March.
    2. Bazaraa, Mokhtar S. & Sherali, Hanif D., 1981. "On the choice of step size in subgradient optimization," European Journal of Operational Research, Elsevier, vol. 7(4), pages 380-388, August.
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    Cited by:

    1. J. Y. Bello Cruz & R. Díaz Millán, 2016. "A relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces," Journal of Global Optimization, Springer, vol. 65(3), pages 597-614, July.
    2. Yunmei Chen & Xiaojing Ye & Wei Zhang, 2020. "Acceleration techniques for level bundle methods in weakly smooth convex constrained optimization," Computational Optimization and Applications, Springer, vol. 77(2), pages 411-432, November.
    3. Wim Ackooij & Welington Oliveira, 2019. "Nonsmooth and Nonconvex Optimization via Approximate Difference-of-Convex Decompositions," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 49-80, July.

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