IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v143y2009i1d10.1007_s10957-009-9551-2.html
   My bibliography  Save this article

Dini Set-Valued Directional Derivative in Locally Lipschitz Vector Optimization

Author

Listed:
  • I. Ginchev

    (University of Insubria)

  • A. Guerraggio

    (University of Insubria)

  • M. Rocca

    (University of Insubria)

Abstract

The present paper studies the following constrained vector optimization problem: min C f(x), g(x)∈−K, h(x)=0, where f:ℝ n →ℝ m , g:ℝ n →ℝ p and h:ℝ n →ℝ q are locally Lipschitz functions and C⊂ℝ m , K⊂ℝ p are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient conditions are obtained for a point x 0 to be a w-minimizer (weakly efficient point) or an i-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven. Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo.

Suggested Citation

  • I. Ginchev & A. Guerraggio & M. Rocca, 2009. "Dini Set-Valued Directional Derivative in Locally Lipschitz Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 87-105, October.
    • Handle: RePEc:spr:joptap:v:143:y:2009:i:1:d:10.1007_s10957-009-9551-2
    DOI: 10.1007/s10957-009-9551-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-009-9551-2
    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/s10957-009-9551-2?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. P. Q. Khanh & N. D. Tuan, 2007. "Optimality Conditions for Nonsmooth Multiobjective Optimization Using Hadamard Directional Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 341-357, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Thai Doan Chuong, 2019. "Optimality and Duality in Nonsmooth Conic Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 471-489, November.
    2. Elena Constantin, 2019. "Necessary conditions for weak efficiency for nonsmooth degenerate multiobjective optimization problems," Journal of Global Optimization, Springer, vol. 75(1), pages 111-129, September.

    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. P. Q. Khanh & N. M. Tung, 2015. "Second-Order Optimality Conditions with the Envelope-Like Effect for Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 68-90, October.
    2. Nguyen Minh Tung & Nguyen Xuan Duy Bao, 2023. "New Set-Valued Directional Derivatives: Calculus and Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 411-437, May.
    3. Phan Quoc Khanh & Nguyen Minh Tung, 2016. "Second-Order Conditions for Open-Cone Minimizers and Firm Minimizers in Set-Valued Optimization Subject to Mixed Constraints," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 45-69, October.
    4. Nguyen Minh Tung & Nguyen Xuan Duy Bao, 2022. "Higher-order set-valued Hadamard directional derivatives: calculus rules and sensitivity analysis of equilibrium problems and generalized equations," Journal of Global Optimization, Springer, vol. 83(2), pages 377-402, June.
    5. Luis Rodríguez-Marín & Miguel Sama, 2013. "Scalar Lagrange Multiplier Rules for Set-Valued Problems in Infinite-Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 683-700, March.
    6. Khanh, Phan Quoc & Quyen, Ho Thuc & Yao, Jen-Chih, 2011. "Optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials," European Journal of Operational Research, Elsevier, vol. 212(2), pages 235-241, July.
    7. Ram U. Verma & G. J. Zalmai, 2018. "Parameter-free duality models and applications to semiinfinite minmax fractional programming based on second-order ( $$\phi ,\eta ,\rho ,\theta ,{\tilde{m}}$$ ϕ , η , ρ , θ , m ~ )-sonvexities," OPSEARCH, Springer;Operational Research Society of India, vol. 55(2), pages 381-410, June.
    8. Tuan, Nguyen Dinh, 2015. "First and second-order optimality conditions for nonsmooth vector optimization using set-valued directional derivatives," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 300-317.
    9. Giorgio Giorgi, 2021. "Some Classical Directional Derivatives and Their Use in Optimization," DEM Working Papers Series 204, University of Pavia, Department of Economics and Management.

    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:143:y:2009:i:1:d:10.1007_s10957-009-9551-2. 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.