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Family of Projected Descent Methods for Optimization Problems with Simple Bounds

Author

Listed:
  • A. Schwartz

    (University of California)

  • E. Polak

    (University of California)

Abstract

This paper presents a family of projected descent direction algorithms with inexact line search for solving large-scale minimization problems subject to simple bounds on the decision variables. The global convergence of algorithms in this family is ensured by conditions on the descent directions and line search. Whenever a sequence constructed by an algorithm in this family enters a sufficiently small neighborhood of a local minimizer ○ satisfying standard second-order sufficiency conditions, it gets trapped and converges to this local minimizer. Furthermore, in this case, the active constraint set at ○ is identified in a finite number of iterations. This fact is used to ensure that the rate of convergence to a local minimizer, satisfying standard second-order sufficiency conditions, depends only on the behavior of the algorithm in the unconstrained subspace. As a particular example, we present projected versions of the modified Polak–Ribière conjugate gradient method and the limited-memory BFGS quasi-Newton method that retain the convergence properties associated with those algorithms applied to unconstrained problems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:92:y:1997:i:1:d:10.1023_a:1022690711754
    DOI: 10.1023/A:1022690711754
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    Citations

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    Cited by:

    1. N.H. Xiu & J.Z. Zhang, 2002. "Local Convergence Analysis of Projection-Type Algorithms: Unified Approach," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 211-230, October.
    2. 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.
    3. 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.
    4. 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.
    5. Hoai Le Thi & Van Huynh & Tao Dinh & A. Vaz & L. Vicente, 2014. "Globally convergent DC trust-region methods," Journal of Global Optimization, Springer, vol. 59(2), pages 209-225, July.
    6. David Ek & Anders Forsgren, 2021. "Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization," Computational Optimization and Applications, Springer, vol. 79(1), pages 155-191, May.
    7. Denis Becker & Alexei Gaivoronski, 2014. "Stochastic optimization on social networks with application to service pricing," Computational Management Science, Springer, vol. 11(4), pages 531-562, October.
    8. Johannes O. Royset, 2016. "Preface," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 713-718, June.

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