IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v85y2023i2d10.1007_s10898-022-01217-0.html
   My bibliography  Save this article

An active set strategy to address the ill-conditioning of smoothing methods for solving finite linear minimax problems

Author

Listed:
  • Zhengyong Zhou

    (Shanxi Normal University)

  • Xiaoyang Dai

    (Shanxi Normal University)

Abstract

In this paper, an active set strategy is presented to address the ill-conditioning of smoothing methods for solving finite linear minimax problems. Based on the first order optimality conditions, a concept of the strongly active set composed of a part of active indexes is introduced. In the active set strategy, a strongly active set is obtained by solving a linear system or a linear programming problem, then an optimal solution with its active set and Lagrange multipliers is computed by an iterative process. A hybrid algorithm combining a smoothing algorithm and the active set strategy is proposed for solving finite linear minimax problems, in which an approximate solution is obtained by the smoothing algorithm, then an optimal solution is computed by the active set strategy. The convergences of the active set strategy and the hybrid algorithm are established for general finite linear minimax problems. Preliminary numerical experiments show that the active set strategy and the hybrid algorithm are effective and robust, and the active set strategy can effectively address the ill-conditioning of smoothing methods for solving general finite linear minimax problems.

Suggested Citation

  • Zhengyong Zhou & Xiaoyang Dai, 2023. "An active set strategy to address the ill-conditioning of smoothing methods for solving finite linear minimax problems," Journal of Global Optimization, Springer, vol. 85(2), pages 421-439, February.
    • Handle: RePEc:spr:jglopt:v:85:y:2023:i:2:d:10.1007_s10898-022-01217-0
    DOI: 10.1007/s10898-022-01217-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-022-01217-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-022-01217-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. E. Obasanjo & G. Tzallas-Regas & B. Rustem, 2010. "An Interior-Point Algorithm for Nonlinear Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 291-318, February.
    2. Pierre Hansen & Dominique Peeters & Denis Richard & Jacques-Francois Thisse, 1985. "The Minisum and Minimax Location Problems Revisited," Operations Research, INFORMS, vol. 33(6), pages 1251-1265, December.
    3. E. Polak & R. S. Womersley & H. X. Yin, 2008. "An Algorithm Based on Active Sets and Smoothing for Discretized Semi-Infinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 311-328, August.
    4. E. Polak & J. O. Royset & R. S. Womersley, 2003. "Algorithms with Adaptive Smoothing for Finite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 459-484, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. E. Y. Pee & J. O. Royset, 2011. "On Solving Large-Scale Finite Minimax Problems Using Exponential Smoothing," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 390-421, February.
    2. Junxiang Li & Mingsong Cheng & Bo Yu & Shuting Zhang, 2015. "Group Update Method for Sparse Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 257-277, July.
    3. J. O. Royset & E. Y. Pee, 2012. "Rate of Convergence Analysis of Discretization and Smoothing Algorithms for Semiinfinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 855-882, December.
    4. Jin-bao Jian & Xing-de Mo & Li-juan Qiu & Su-ming Yang & Fu-sheng Wang, 2014. "Simple Sequential Quadratically Constrained Quadratic Programming Feasible Algorithm with Active Identification Sets for Constrained Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 160(1), pages 158-188, January.
    5. Rockafellar, R.T. & Royset, J.O., 2010. "On buffered failure probability in design and optimization of structures," Reliability Engineering and System Safety, Elsevier, vol. 95(5), pages 499-510.
    6. J. Redondo & J. Fernández & I. García & P. Ortigosa, 2009. "A robust and efficient algorithm for planar competitive location problems," Annals of Operations Research, Springer, vol. 167(1), pages 87-105, March.
    7. B. Rustem & S. Žaković & P. Parpas, 2008. "Convergence of an Interior Point Algorithm for Continuous Minimax," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 87-103, January.
    8. Romero-Morales, Dolores & Carrizosa, Emilio & Conde, Eduardo, 1997. "Semi-obnoxious location models: A global optimization approach," European Journal of Operational Research, Elsevier, vol. 102(2), pages 295-301, October.
    9. Rafael Blanquero & Emilio Carrizosa & Amaya Nogales-Gómez & Frank Plastria, 2014. "Single-facility huff location problems on networks," Annals of Operations Research, Springer, vol. 222(1), pages 175-195, November.
    10. W. Hare & J. Nutini, 2013. "A derivative-free approximate gradient sampling algorithm for finite minimax problems," Computational Optimization and Applications, Springer, vol. 56(1), pages 1-38, September.
    11. Wei-jie Cong & Le Wang & Hui Sun, 2020. "Rank-two update algorithm versus Frank–Wolfe algorithm with away steps for the weighted Euclidean one-center problem," Computational Optimization and Applications, Springer, vol. 75(1), pages 237-262, January.
    12. E. Polak & R. S. Womersley & H. X. Yin, 2008. "An Algorithm Based on Active Sets and Smoothing for Discretized Semi-Infinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 311-328, August.
    13. Daniel Scholz, 2010. "The multicriteria big cube small cube method," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(1), pages 286-302, July.
    14. Carrizosa, E. & Frenk, J.B.G., 1996. "Dominating Sets for Convex Functions with some Applications," Econometric Institute Research Papers EI 9657-/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    15. Klamroth, K., 2004. "Algebraic properties of location problems with one circular barrier," European Journal of Operational Research, Elsevier, vol. 154(1), pages 20-35, April.
    16. Skriver, Anders J. V. & Andersen, Kim Allan, 2003. "The bicriterion semi-obnoxious location (BSL) problem solved by an [epsilon]-approximation," European Journal of Operational Research, Elsevier, vol. 146(3), pages 517-528, May.
    17. Avella, P. & Benati, S. & Canovas Martinez, L. & Dalby, K. & Di Girolamo, D. & Dimitrijevic, B. & Ghiani, G. & Giannikos, I. & Guttmann, N. & Hultberg, T. H. & Fliege, J. & Marin, A. & Munoz Marquez, , 1998. "Some personal views on the current state and the future of locational analysis," European Journal of Operational Research, Elsevier, vol. 104(2), pages 269-287, January.
    18. M. Hakan Akyüz & Temel Öncan & İ. Kuban Altınel, 2019. "Branch and bound algorithms for solving the multi-commodity capacitated multi-facility Weber problem," Annals of Operations Research, Springer, vol. 279(1), pages 1-42, August.
    19. Fernandez, J. & Fernandez, P. & Pelegrin, B., 2000. "A continuous location model for siting a non-noxious undesirable facility within a geographical region," European Journal of Operational Research, Elsevier, vol. 121(2), pages 259-274, March.
    20. Stolletz, Raik & Stolletz, Lars, 2008. "Linearisierungsverfahren für Standortplanungsprobleme mit nichtlinearen Transportkosten," Hannover Economic Papers (HEP) dp-388, Leibniz Universität Hannover, Wirtschaftswissenschaftliche Fakultät.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:85:y:2023:i:2:d:10.1007_s10898-022-01217-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.