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A Two-Stage Active-Set Algorithm for Bound-Constrained Optimization

Author

Listed:
  • Andrea Cristofari

    (Sapienza University of Rome)

  • Marianna Santis

    (Alpen-Adria-Universität Klagenfurt)

  • Stefano Lucidi

    (Sapienza University of Rome)

  • Francesco Rinaldi

    (University of Padova)

Abstract

In this paper, we describe a two-stage method for solving optimization problems with bound constraints. It combines the active-set estimate described in Facchinei and Lucidi (J Optim Theory Appl 85(2):265–289, 1995) with a modification of the non-monotone line search framework recently proposed in De Santis et al. (Comput Optim Appl 53(2):395–423, 2012). In the first stage, the algorithm exploits a property of the active-set estimate that ensures a significant reduction in the objective function when setting to the bounds all those variables estimated active. In the second stage, a truncated-Newton strategy is used in the subspace of the variables estimated non-active. In order to properly combine the two phases, a proximity check is included in the scheme. This new tool, together with the other theoretical features of the two stages, enables us to prove global convergence. Furthermore, under additional standard assumptions, we can show that the algorithm converges at a superlinear rate. Promising experimental results demonstrate the effectiveness of the proposed method.

Suggested Citation

  • Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2017. "A Two-Stage Active-Set Algorithm for Bound-Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 369-401, February.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:2:d:10.1007_s10957-016-1024-9
    DOI: 10.1007/s10957-016-1024-9
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    References listed on IDEAS

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    1. Wanyou Cheng & Donghui Li, 2012. "An Active Set Modified Polak–Ribiére–Polyak Method for Large-Scale Nonlinear Bound Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 1084-1094, December.
    2. A. Schwartz & E. Polak, 1997. "Family of Projected Descent Methods for Optimization Problems with Simple Bounds," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 1-31, January.
    3. Nicholas Gould & Dominique Orban & Philippe Toint, 2015. "CUTEst: a Constrained and Unconstrained Testing Environment with safe threads for mathematical optimization," Computational Optimization and Applications, Springer, vol. 60(3), pages 545-557, April.
    4. M. Santis & G. Pillo & S. Lucidi, 2012. "An active set feasible method for large-scale minimization problems with bound constraints," Computational Optimization and Applications, Springer, vol. 53(2), pages 395-423, October.
    5. Ernesto Birgin & Jan Gentil, 2012. "Evaluating bound-constrained minimization software," Computational Optimization and Applications, Springer, vol. 53(2), pages 347-373, October.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2022. "Minimization over the $$\ell _1$$ ℓ 1 -ball using an active-set non-monotone projected gradient," Computational Optimization and Applications, Springer, vol. 83(2), pages 693-721, November.
    2. Cristofari, Andrea, 2023. "A decomposition method for lasso problems with zero-sum constraint," European Journal of Operational Research, Elsevier, vol. 306(1), pages 358-369.
    3. Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2020. "An active-set algorithmic framework for non-convex optimization problems over the simplex," Computational Optimization and Applications, Springer, vol. 77(1), pages 57-89, September.
    4. Enrico Bettiol & Lucas Létocart & Francesco Rinaldi & Emiliano Traversi, 2020. "A conjugate direction based simplicial decomposition framework for solving a specific class of dense convex quadratic programs," Computational Optimization and Applications, Springer, vol. 75(2), pages 321-360, March.
    5. Andrea Cristofari & Gianni Di Pillo & Giampaolo Liuzzi & Stefano Lucidi, 2022. "An Augmented Lagrangian Method Exploiting an Active-Set Strategy and Second-Order Information," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 300-323, June.

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