IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v193y2022i1d10.1007_s10957-022-02019-w.html
   My bibliography  Save this article

Solution of Fractional Quadratic Programs on the Simplex and Application to the Eigenvalue Complementarity Problem

Author

Listed:
  • Joaquim Júdice

    (FCTUC - DEEC Polo II Universidade de Coimbra)

  • Valentina Sessa

    (PSL Research University)

  • Masao Fukushima

    (The Kyoto College of Graduate Studies for Informatics)

Abstract

In this paper, we introduce an implementation of Dinkelbach’s algorithm for computing a global maximum of a fractional linear quadratic program (FLQP) on the simplex that employs an efficient block principal pivoting algorithm in each iteration. A new sequential FLQP algorithm is introduced for computing a stationary point (SP) of a fractional quadratic program (FQP) on the simplex. Global convergence for this algorithm is established. This sequential algorithm is recommended for the solution of the symmetric eigenvalue complementarity problem (EiCP), as this problem is equivalent to the computation of an SP of an FQP on the simplex. Computational experience reported in this paper indicates that the implementation of Dinkelbach’s method for the FLQP and the sequential FLQP algorithm are quite efficient in practice. An extension of the sequential FLQP algorithm for solving the nonsymmetric EiCP is also introduced. Since this method solves a special variational inequality (VI) problem in each iteration, it can be considered as a sequential VI algorithm. Although the convergence of this algorithm has yet to be established, preliminary computational experience indicates that the sequential VI algorithm is quite a promising technique for the solution of the nonsymmetric EiCP.

Suggested Citation

  • Joaquim Júdice & Valentina Sessa & Masao Fukushima, 2022. "Solution of Fractional Quadratic Programs on the Simplex and Application to the Eigenvalue Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 545-573, June.
    • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-022-02019-w
    DOI: 10.1007/s10957-022-02019-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-022-02019-w
    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/s10957-022-02019-w?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. Luís Fernandes & Joaquim Júdice & Hanif Sherali & Masao Fukushima, 2014. "On the computation of all eigenvalues for the eigenvalue complementarity problem," Journal of Global Optimization, Springer, vol. 59(2), pages 307-326, July.
    2. 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.
    3. Siegfried Schaible, 1976. "Fractional Programming. II, On Dinkelbach's Algorithm," Management Science, INFORMS, vol. 22(8), pages 868-873, April.
    4. Masao Fukushima & Joaquim Júdice & Welington Oliveira & Valentina Sessa, 2020. "A sequential partial linearization algorithm for the symmetric eigenvalue complementarity problem," Computational Optimization and Applications, Springer, vol. 77(3), pages 711-728, December.
    5. Schaible, Siegfried, 1981. "Fractional programming: Applications and algorithms," European Journal of Operational Research, Elsevier, vol. 7(2), pages 111-120, June.
    6. Werner Dinkelbach, 1967. "On Nonlinear Fractional Programming," Management Science, INFORMS, vol. 13(7), pages 492-498, March.
    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. Cook, Wade D. & Zhu, Joe, 2007. "Within-group common weights in DEA: An analysis of power plant efficiency," European Journal of Operational Research, Elsevier, vol. 178(1), pages 207-216, April.
    2. Niu, Yi-Shuai & Júdice, Joaquim & Le Thi, Hoai An & Pham, Dinh Tao, 2019. "Improved dc programming approaches for solving the quadratic eigenvalue complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 95-113.
    3. R. Yamamoto & H. Konno, 2007. "An Efficient Algorithm for Solving Convex–Convex Quadratic Fractional Programs," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 241-255, May.
    4. Meijia Yang & Yong Xia & Jiulin Wang & Jiming Peng, 2018. "Efficiently solving total least squares with Tikhonov identical regularization," Computational Optimization and Applications, Springer, vol. 70(2), pages 571-592, June.
    5. Harold P. Benson, 2006. "Maximizing the ratio of two convex functions over a convex set," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(4), pages 309-317, June.
    6. Juan S. Borrero & Colin Gillen & Oleg A. Prokopyev, 2017. "Fractional 0–1 programming: applications and algorithms," Journal of Global Optimization, Springer, vol. 69(1), pages 255-282, September.
    7. Abderrahman Bouhamidi & Mohammed Bellalij & Rentsen Enkhbat & Khalid Jbilou & Marcos Raydan, 2018. "Conditional Gradient Method for Double-Convex Fractional Programming Matrix Problems," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 163-177, January.
    8. Bram L. Gorissen, 2015. "Robust Fractional Programming," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 508-528, August.
    9. Wassila Drici & Fatma Zohra Ouail & Mustapha Moulaï, 2018. "Optimizing a linear fractional function over the integer efficient set," Annals of Operations Research, Springer, vol. 267(1), pages 135-151, August.
    10. Wong, Man Hong, 2013. "Investment models based on clustered scenario trees," European Journal of Operational Research, Elsevier, vol. 227(2), pages 314-324.
    11. Benson, Harold P., 2006. "Fractional programming with convex quadratic forms and functions," European Journal of Operational Research, Elsevier, vol. 173(2), pages 351-369, September.
    12. Garrido, Rodrigo A. & Bronfman, Andrés C., 2017. "Equity and social acceptability in multiple hazardous materials routing through urban areas," Transportation Research Part A: Policy and Practice, Elsevier, vol. 102(C), pages 244-260.
    13. Xiang-Kai Sun & Xian-Jun Long & Yi Chai, 2015. "Sequential Optimality Conditions for Fractional Optimization with Applications to Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 479-499, February.
    14. Tunjo Perić & Josip Matejaš & Zoran Babić, 2023. "Advantages, sensitivity and application efficiency of the new iterative method to solve multi-objective linear fractional programming problem," 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. 31(3), pages 751-767, September.
    15. Luca Consolini & Marco Locatelli & Jiulin Wang & Yong Xia, 2020. "Efficient local search procedures for quadratic fractional programming problems," Computational Optimization and Applications, Springer, vol. 76(1), pages 201-232, May.
    16. Harald Dyckhoff & Katrin Allen, 1999. "Theoretische Begründung einer Effizienzanalyse mittels Data Envelopment Analysis (DEA)," Schmalenbach Journal of Business Research, Springer, vol. 51(5), pages 411-436, May.
    17. Feng Guo & Liguo Jiao, 2023. "A new scheme for approximating the weakly efficient solution set of vector rational optimization problems," Journal of Global Optimization, Springer, vol. 86(4), pages 905-930, August.
    18. Maziar Sahamkhadam, 2021. "Dynamic copula-based expectile portfolios," Journal of Asset Management, Palgrave Macmillan, vol. 22(3), pages 209-223, May.
    19. Laurent Alfandari & Alborz Hassanzadeh & Ivana Ljubić, 2021. "An Exact Method for Assortment Optimization under the Nested Logit Model," Working Papers hal-02463159, HAL.
    20. A. Roubi, 2000. "Method of Centers for Generalized Fractional Programming," Journal of Optimization Theory and Applications, Springer, vol. 107(1), pages 123-143, October.

    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:joptap:v:193:y:2022:i:1:d:10.1007_s10957-022-02019-w. 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.