IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v83y2022i2d10.1007_s10898-021-01110-2.html
   My bibliography  Save this article

Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces

Author

Listed:
  • Regina S. Burachik

    (UniSA STEM, University of South Australia)

  • Yaohua Hu

    (Shenzhen University)

  • Xiaoqi Yang

    (The Hong Kong Polytechnic University)

Abstract

An interior quasi-subgradient method is proposed based on the proximal distance to solve constrained nondifferentiable quasi-convex optimization problems in Hilbert spaces. It is shown that a newly introduced generalized Gâteaux subdifferential is a subset of a quasi-subdifferential. The convergence properties, including the global convergence and iteration complexity, are investigated under the assumption of the Hölder condition of order p, when using the constant/diminishing/dynamic stepsize rules. Convergence rate results are obtained by assuming a Hölder-type weak sharp minimum condition relative to an induced proximal distance.

Suggested Citation

  • Regina S. Burachik & Yaohua Hu & Xiaoqi Yang, 2022. "Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces," Journal of Global Optimization, Springer, vol. 83(2), pages 249-271, June.
    • Handle: RePEc:spr:jglopt:v:83:y:2022:i:2:d:10.1007_s10898-021-01110-2
    DOI: 10.1007/s10898-021-01110-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-021-01110-2
    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/s10898-021-01110-2?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. Alfred Auslender & Marc Teboulle & Sami Ben-Tiba, 1999. "Interior Proximal and Multiplier Methods Based on Second Order Homogeneous Kernels," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 645-668, August.
    2. Yaohua Hu & Jiawen Li & Carisa Kwok Wai Yu, 2020. "Convergence rates of subgradient methods for quasi-convex optimization problems," Computational Optimization and Applications, Springer, vol. 77(1), pages 183-212, September.
    3. Souza, Sissy da S. & Oliveira, P.R. & da Cruz Neto, J.X. & Soubeyran, A., 2010. "A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant," European Journal of Operational Research, Elsevier, vol. 201(2), pages 365-376, March.
    4. Kengy Barty & Jean-Sébastien Roy & Cyrille Strugarek, 2007. "Hilbert-Valued Perturbed Subgradient Algorithms," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 551-562, August.
    5. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
    6. Yaohua Hu & Carisa Kwok Wai Yu & Xiaoqi Yang, 2019. "Incremental quasi-subgradient methods for minimizing the sum of quasi-convex functions," Journal of Global Optimization, Springer, vol. 75(4), pages 1003-1028, December.
    7. Papa Quiroz, E.A. & Roberto Oliveira, P., 2012. "An extension of proximal methods for quasiconvex minimization on the nonnegative orthant," European Journal of Operational Research, Elsevier, vol. 216(1), pages 26-32.
    8. Papa Quiroz, E.A. & Mallma Ramirez, L. & Oliveira, P.R., 2015. "An inexact proximal method for quasiconvex minimization," European Journal of Operational Research, Elsevier, vol. 246(3), pages 721-729.
    9. M. V. Solodov & S. K. Zavriev, 1998. "Error Stability Properties of Generalized Gradient-Type Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 663-680, September.
    10. Hu, Yaohua & Yang, Xiaoqi & Sim, Chee-Khian, 2015. "Inexact subgradient methods for quasi-convex optimization problems," European Journal of Operational Research, Elsevier, vol. 240(2), pages 315-327.
    11. Arnaldo S. Brito & J. X. Cruz Neto & Jurandir O. Lopes & P. Roberto Oliveira, 2012. "Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 217-234, July.
    12. A. Auslender & M. Teboulle, 2004. "Interior Gradient and Epsilon-Subgradient Descent Methods for Constrained Convex Minimization," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 1-26, February.
    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. Hu, Yaohua & Li, Gongnong & Yu, Carisa Kwok Wai & Yip, Tsz Leung, 2022. "Quasi-convex feasibility problems: Subgradient methods and convergence rates," European Journal of Operational Research, Elsevier, vol. 298(1), pages 45-58.
    2. E. A. Papa Quiroz & S. Cruzado, 2022. "An inexact scalarization proximal point method for multiobjective quasiconvex minimization," Annals of Operations Research, Springer, vol. 316(2), pages 1445-1470, September.
    3. Papa Quiroz, E.A. & Mallma Ramirez, L. & Oliveira, P.R., 2015. "An inexact proximal method for quasiconvex minimization," European Journal of Operational Research, Elsevier, vol. 246(3), pages 721-729.
    4. Yaohua Hu & Jiawen Li & Carisa Kwok Wai Yu, 2020. "Convergence rates of subgradient methods for quasi-convex optimization problems," Computational Optimization and Applications, Springer, vol. 77(1), pages 183-212, September.
    5. H. Apolinário & E. Papa Quiroz & P. Oliveira, 2016. "A scalarization proximal point method for quasiconvex multiobjective minimization," Journal of Global Optimization, Springer, vol. 64(1), pages 79-96, January.
    6. Arnaldo S. Brito & J. X. Cruz Neto & Jurandir O. Lopes & P. Roberto Oliveira, 2012. "Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 217-234, July.
    7. F. Lara, 2022. "On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 891-911, March.
    8. Papa Quiroz, E.A. & Roberto Oliveira, P., 2012. "An extension of proximal methods for quasiconvex minimization on the nonnegative orthant," European Journal of Operational Research, Elsevier, vol. 216(1), pages 26-32.
    9. Xiaoqi Yang & Chenchen Zu, 2022. "Convergence of Inexact Quasisubgradient Methods with Extrapolation," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 676-703, June.
    10. Yen-Huan Li & Volkan Cevher, 2019. "Convergence of the Exponentiated Gradient Method with Armijo Line Search," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 588-607, May.
    11. Shota Takahashi & Mituhiro Fukuda & Mirai Tanaka, 2022. "New Bregman proximal type algorithms for solving DC optimization problems," Computational Optimization and Applications, Springer, vol. 83(3), pages 893-931, December.
    12. Shinji Yamada & Akiko Takeda, 2018. "Successive Lagrangian relaxation algorithm for nonconvex quadratic optimization," Journal of Global Optimization, Springer, vol. 71(2), pages 313-339, June.
    13. Zamani, Moslem & Abbaszadehpeivasti, Hadi & de Klerk, Etienne, 2023. "The exact worst-case convergence rate of the alternating direction method of multipliers," Other publications TiSEM f30ae9e6-ed19-423f-bd1e-0, Tilburg University, School of Economics and Management.
    14. Larsson, Torbjorn & Patriksson, Michael & Stromberg, Ann-Brith, 2003. "On the convergence of conditional [var epsilon]-subgradient methods for convex programs and convex-concave saddle-point problems," European Journal of Operational Research, Elsevier, vol. 151(3), pages 461-473, December.
    15. Sorin-Mihai Grad & Felipe Lara, 2022. "An extension of the proximal point algorithm beyond convexity," Journal of Global Optimization, Springer, vol. 82(2), pages 313-329, February.
    16. Elena Tovbis & Vladimir Krutikov & Predrag Stanimirović & Vladimir Meshechkin & Aleksey Popov & Lev Kazakovtsev, 2023. "A Family of Multi-Step Subgradient Minimization Methods," Mathematics, MDPI, vol. 11(10), pages 1-24, May.
    17. J. X. Cruz Neto & P. R. Oliveira & A. Soubeyran & J. C. O. Souza, 2020. "A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem," Annals of Operations Research, Springer, vol. 289(2), pages 313-339, June.
    18. Yaohua Hu & Carisa Kwok Wai Yu & Xiaoqi Yang, 2019. "Incremental quasi-subgradient methods for minimizing the sum of quasi-convex functions," Journal of Global Optimization, Springer, vol. 75(4), pages 1003-1028, December.
    19. Regina Sandra Burachik & B. F. Svaiter, 2001. "A Relative Error Tolerance for a Family of Generalized Proximal Point Methods," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 816-831, November.
    20. HyungSeon Oh, 2021. "Distributed optimal power flow," PLOS ONE, Public Library of Science, vol. 16(6), pages 1-27, June.

    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:83:y:2022:i:2:d:10.1007_s10898-021-01110-2. 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.