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An active-set algorithmic framework for non-convex optimization problems over the simplex

Author

Listed:
  • Andrea Cristofari

    (Università di Padova)

  • Marianna Santis

    (Sapienza Università di Roma)

  • Stefano Lucidi

    (Sapienza Università di Roma)

  • Francesco Rinaldi

    (Università di Padova)

Abstract

In this paper, we describe a new active-set algorithmic framework for minimizing a non-convex function over the unit simplex. At each iteration, the method makes use of a rule for identifying active variables (i.e., variables that are zero at a stationary point) and specific directions (that we name active-set gradient related directions) satisfying a new “nonorthogonality” type of condition. We prove global convergence to stationary points when using an Armijo line search in the given framework. We further describe three different examples of active-set gradient related directions that guarantee linear convergence rate (under suitable assumptions). Finally, we report numerical experiments showing the effectiveness of the approach.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:77:y:2020:i:1:d:10.1007_s10589-020-00195-x
    DOI: 10.1007/s10589-020-00195-x
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    References listed on IDEAS

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    1. de Klerk, E., 2006. "The Complexity of Optimizing over a Simplex, Hypercube or Sphere : A Short Survey," Discussion Paper 2006-85, Tilburg University, Center for Economic Research.
    2. L. Fernandes & A. Fischer & J. Júdice & C. Requejo & J. Soares, 1998. "A block active set algorithm for large-scalequadratic programming with box constraints," Annals of Operations Research, Springer, vol. 81(0), pages 75-96, June.
    3. Brás, Carmo P. & Fischer, Andreas & Júdice, Joaquim J. & Schönefeld, Klaus & Seifert, Sarah, 2017. "A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 36-48.
    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. de Klerk, E., 2008. "The complexity of optimizing over a simplex, hypercube or sphere : A short survey," Other publications TiSEM 485b6860-cf1d-4cad-97b8-2, Tilburg University, School of Economics and Management.
    6. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    7. Etienne Klerk, 2008. "The complexity of optimizing over a simplex, hypercube or sphere: a short survey," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(2), pages 111-125, June.
    8. 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.
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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. Francesco Rinaldi & Damiano Zeffiro, 2023. "Avoiding bad steps in Frank-Wolfe variants," Computational Optimization and Applications, Springer, vol. 84(1), pages 225-264, January.
    4. Immanuel M. Bomze & Francesco Rinaldi & Damiano Zeffiro, 2021. "Frank–Wolfe and friends: a journey into projection-free first-order optimization methods," 4OR, Springer, vol. 19(3), pages 313-345, September.

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