IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v201y2024i2d10.1007_s10957-024-02424-3.html
   My bibliography  Save this article

Computing Critical Angles Between Two Convex Cones

Author

Listed:
  • Welington Oliveira

    (Mines Paris, Université PSL)

  • Valentina Sessa

    (Mines Paris, Université PSL)

  • David Sossa

    (Universidad de O’Higgins)

Abstract

This paper addresses the numerical computation of critical angles between two convex cones in Euclidean spaces. We present a novel approach to computing these critical angles by reducing the problem to finding stationary points of a fractional programming problem. To efficiently compute these stationary points, we introduce a partial linearization-like algorithm that offers significant computational advantages. Solving a sequence of strictly convex subproblems with straightforward solutions in several settings gives the proposed algorithm high computational efficiency while delivering reliable results: our theoretical analysis demonstrates that the proposed algorithm asymptotically computes critical angles. Numerical experiments validate the efficiency of our approach, even when dealing with problems of relatively large dimensions: only a few seconds are necessary to compute critical angles between different types of cones in spaces of dimension 1000.

Suggested Citation

  • Welington Oliveira & Valentina Sessa & David Sossa, 2024. "Computing Critical Angles Between Two Convex Cones," Journal of Optimization Theory and Applications, Springer, vol. 201(2), pages 866-898, May.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:2:d:10.1007_s10957-024-02424-3
    DOI: 10.1007/s10957-024-02424-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02424-3
    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/s10957-024-02424-3?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. Jochen Gorski & Frank Pfeuffer & Kathrin Klamroth, 2007. "Biconvex sets and optimization with biconvex functions: a survey and extensions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(3), pages 373-407, December.
    2. Alberto Seeger & David Sossa, 2015. "Complementarity problems with respect to Loewnerian cones," Journal of Global Optimization, Springer, vol. 62(2), pages 299-318, June.
    3. Siegfried Schaible, 1976. "Fractional Programming. II, On Dinkelbach's Algorithm," Management Science, INFORMS, vol. 22(8), pages 868-873, April.
    4. Gábor Pataki, 2007. "On the Closedness of the Linear Image of a Closed Convex Cone," Mathematics of Operations Research, INFORMS, vol. 32(2), pages 395-412, May.
    5. 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.
    6. 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.
    7. Radu Ioan Boţ & Minh N. Dao & Guoyin Li, 2022. "Extrapolated Proximal Subgradient Algorithms for Nonconvex and Nonsmooth Fractional Programs," Mathematics of Operations Research, INFORMS, vol. 47(3), pages 2415-2443, August.
    8. Héctor Ramírez & David Sossa, 2017. "On the Central Paths in Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 649-668, February.
    9. Schaible, Siegfried, 1981. "Fractional programming: Applications and algorithms," European Journal of Operational Research, Elsevier, vol. 7(2), pages 111-120, June.
    10. 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. 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.
    2. 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.
    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. Tajbakhsh, Alireza & Hassini, Elkafi, 2018. "Evaluating sustainability performance in fossil-fuel power plants using a two-stage data envelopment analysis," Energy Economics, Elsevier, vol. 74(C), pages 154-178.
    10. 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.
    11. Wong, Man Hong, 2013. "Investment models based on clustered scenario trees," European Journal of Operational Research, Elsevier, vol. 227(2), pages 314-324.
    12. 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.
    13. 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.
    14. M. Barkhagen & S. García & J. Gondzio & J. Kalcsics & J. Kroeske & S. Sabanis & A. Staal, 2023. "Optimising portfolio diversification and dimensionality," Journal of Global Optimization, Springer, vol. 85(1), pages 185-234, January.
    15. 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.
    16. 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.
    17. Lin, Yun Hui & Wang, Yuan & He, Dongdong & Lee, Loo Hay, 2020. "Last-mile delivery: Optimal locker location under multinomial logit choice model," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 142(C).
    18. 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.
    19. 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.
    20. 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.

    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:201:y:2024:i:2:d:10.1007_s10957-024-02424-3. 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.