IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v251y2015icp300-317.html
   My bibliography  Save this article

First and second-order optimality conditions for nonsmooth vector optimization using set-valued directional derivatives

Author

Listed:
  • Tuan, Nguyen Dinh

Abstract

We investigate a nonsmooth vector optimization problem with a feasible set defined by a generalized inequality constraint, an equality constraint and a set constraint. Both necessary and sufficient optimality conditions of first and second-order for weak solutions and firm solutions are established in terms of Fritz-John–Lagrange multiplier rules using set-valued directional derivatives and tangent cones and second-order tangent sets. We impose steadiness and strict differentiability for first and second-order necessary conditions, respectively; stability and l-stability for first and second-order sufficient conditions, respectively. The obtained results improve or include some recent known ones. Several illustrative examples are also provided.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:251:y:2015:i:c:p:300-317
    DOI: 10.1016/j.amc.2014.11.061
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300314015938
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2014.11.061?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.
    2. Bienvenido Jiménez & Vicente Novo, 2003. "Second order necessary conditions in set constrained differentiable vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(2), pages 299-317, November.
    3. Yukihiro Maruyama, 1990. "Second-Order Necessary Conditions for Nonlinear Optimization Problems in Banach Spaces and Their Application to an Optimal Control Problem," Mathematics of Operations Research, INFORMS, vol. 15(3), pages 467-482, August.
    4. V. Jeyakumar & D.T. LUC, 2008. "Nonsmooth Vector Functions and Continuous Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-73717-1, 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. Thinh, Vo Duc & Chuong, Thai Doan & Le Hoang Anh, Nguyen, 2023. "Formulas of first-ordered and second-ordered generalization differentials for convex robust systems with applications," Applied Mathematics and Computation, Elsevier, vol. 455(C).

    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. 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.
    2. María C. Maciel & Sandra A. Santos & Graciela N. Sottosanto, 2011. "On Second-Order Optimality Conditions for Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 332-351, May.
    3. Giorgio Giorgi, 2019. "Notes on Constraint Qualifications for Second-Order Optimality Conditions," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(5), pages 16-32, October.
    4. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department of Economics and Management.
    5. Stephan Dempe & Maria Pilecka, 2015. "Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming," Journal of Global Optimization, Springer, vol. 61(4), pages 769-788, April.
    6. 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.
    7. P. Q. Khanh & N. D. Tuan, 2008. "Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 243-261, November.
    8. 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.
    9. Bienvenido Jiménez & Vicente Novo, 2008. "Higher-order optimality conditions for strict local minima," Annals of Operations Research, Springer, vol. 157(1), pages 183-192, January.
    10. Qamrul Hasan Ansari & Mahboubeh Rezaei, 2012. "Invariant Pseudolinearity with Applications," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 587-601, June.
    11. J. Jahn & A. A. Khan & P. Zeilinger, 2005. "Second-Order Optimality Conditions in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 331-347, May.
    12. 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.
    13. Vaithilingam Jeyakumar & Guoyin Li, 2017. "Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone Programs," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 156-178, January.
    14. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    15. Giuseppe Caristi & Massimiliano Ferrara, 2017. "Necessary conditions for nonsmooth multiobjective semi-infinite problems using Michel–Penot subdifferential," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 40(1), pages 103-113, November.
    16. C. Gutiérrez & B. Jiménez & V. Novo, 2009. "New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 85-106, July.
    17. 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.
    18. Bui Trong Kien & Trinh Duy Binh, 2023. "On the second-order optimality conditions for multi-objective optimal control problems with mixed pointwise constraints," Journal of Global Optimization, Springer, vol. 85(1), pages 155-183, January.
    19. 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.
    20. Alireza Kabgani & Majid Soleimani-damaneh & Moslem Zamani, 2017. "Optimality conditions in optimization problems with convex feasible set using convexificators," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 103-121, 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:eee:apmaco:v:251:y:2015:i:c:p:300-317. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.