IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v194y2022i2d10.1007_s10957-022-02033-y.html
   My bibliography  Save this article

An Invariant-Point Theorem in Banach Space with Applications to Nonconvex Optimization

Author

Listed:
  • Vo Si Trong Long

    (University of Science
    Vietnam National University)

Abstract

An invariant-point theorem and its equivalent formulation are established in which distance functions are replaced by minimal time functions. It is worth emphasizing here that the class of minimal time functions can be interpreted as a general type of directional distance functions recently used to develop new applications in optimization theory. The obtained results are applied in two directions. First, we derive sufficient conditions for the existence of solutions to optimization-related problems without convexity. As an easy corollary, we get a directional Ekeland variational principle. Second, we propose a new type of global error bounds for inequalities which allows us to simultaneously study nonconvex and convex functions. Several examples and comparison remarks are included as well to explain advantages of our results with existing ones in the literature.

Suggested Citation

  • Vo Si Trong Long, 2022. "An Invariant-Point Theorem in Banach Space with Applications to Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 440-464, August.
  • Handle: RePEc:spr:joptap:v:194:y:2022:i:2:d:10.1007_s10957-022-02033-y
    DOI: 10.1007/s10957-022-02033-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-022-02033-y
    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/s10957-022-02033-y?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. Teodor Chelmuş & Marius Durea & Elena-Andreea Florea, 2019. "Directional Pareto Efficiency: Concepts and Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 336-365, July.
    2. Phan Khanh & Vo Long, 2014. "Invariant-point theorems and existence of solutions to optimization-related problems," Journal of Global Optimization, Springer, vol. 58(3), pages 545-564, March.
    3. Boris Mordukhovich & Nguyen Nam, 2010. "Limiting subgradients of minimal time functions in Banach spaces," Journal of Global Optimization, Springer, vol. 46(4), pages 615-633, April.
    4. Aram V. Arutyunov & Alexey F. Izmailov, 2006. "Directional Stability Theorem and Directional Metric Regularity," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 526-543, August.
    5. Nguyen Mau Nam & Maria Cristina Villalobos & Nguyen Thai An, 2012. "Minimal Time Functions and the Smallest Intersecting Ball Problem with Unbounded Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 768-791, September.
    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. Mohamed Ait Mansour & Marius Durea & Hassan Riahi, 2022. "Strict directional solutions in vectorial problems: necessary optimality conditions," Journal of Global Optimization, Springer, vol. 82(1), pages 119-138, January.
    2. Sorin-Mihai Grad & Oleg Wilfer, 2019. "A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality," Journal of Global Optimization, Springer, vol. 74(1), pages 121-160, May.
    3. Andreas Fischer & Alexey F. Izmailov & Mario Jelitte, 2023. "Stability of Singular Solutions of Nonlinear Equations with Restricted Smoothness Assumptions," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 1008-1035, March.
    4. Marius Durea & Diana Maxim & Radu Strugariu, 2021. "Metric Inequality Conditions on Sets and Consequences in Optimization," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 744-771, June.
    5. Boris Mordukhovich & Nguyen Mau Nam, 2011. "Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem," Journal of Optimization Theory and Applications, Springer, vol. 148(3), pages 431-454, March.
    6. Huynh Van Ngai & Michel Théra, 2015. "Directional Metric Regularity of Multifunctions," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 969-991, October.
    7. Xiaolong Qin & Nguyen Thai An, 2019. "Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets," Computational Optimization and Applications, Springer, vol. 74(3), pages 821-850, December.
    8. Huynh Ngai & Nguyen Huu Tron & Michel Théra, 2016. "Directional Hölder Metric Regularity," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 785-819, December.
    9. Nguyen Mau Nam & Maria Cristina Villalobos & Nguyen Thai An, 2012. "Minimal Time Functions and the Smallest Intersecting Ball Problem with Unbounded Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 768-791, September.

    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:194:y:2022:i:2:d:10.1007_s10957-022-02033-y. 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.