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An active set feasible method for large-scale minimization problems with bound constraints

Author

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  • M. Santis
  • G. Pillo
  • S. Lucidi

Abstract

We are concerned with the solution of the bound constrained minimization problem {minf(x), l≤x≤u}. For the solution of this problem we propose an active set method that combines ideas from projected and nonmonotone Newton-type methods. It is based on an iteration of the form x k+1 =[x k +α k d k ] ♯ , where α k is the steplength, d k is the search direction and [⋅] ♯ is the projection operator on the set [l,u]. At each iteration a new formula to estimate the active set is first employed. Then the components $d_{N}^{k}$ of d k corresponding to the free variables are determined by a truncated Newton method, and the components $d_{A}^{k}$ of d k corresponding to the active variables are computed by a Barzilai-Borwein gradient method. The steplength α k is computed by an adaptation of the nonmonotone stabilization technique proposed in Grippo et al. (Numer. Math. 59:779–805, 1991 ). The method is a feasible one, since it maintains feasibility of the iterates x k , and is well suited for large-scale problems, since it uses matrix-vector products only in the truncated Newton method for computing $d_{N}^{k}$ . We prove the convergence of the method, with superlinear rate under usual additional assumptions. An extensive numerical experimentation performed on an algorithmic implementation shows that the algorithm compares favorably with other widely used codes for bound constrained problems. Copyright Springer Science+Business Media New York 2012

Suggested Citation

  • 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.
  • Handle: RePEc:spr:coopap:v:53:y:2012:i:2:p:395-423
    DOI: 10.1007/s10589-012-9506-7
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. 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.
    2. 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.
    3. Christoph Buchheim & Renke Kuhlmann & Christian Meyer, 2018. "Combinatorial optimal control of semilinear elliptic PDEs," Computational Optimization and Applications, Springer, vol. 70(3), pages 641-675, July.
    4. 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.
    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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