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Approximate Karush–Kuhn–Tucker Condition in Multiobjective Optimization

Author

Listed:
  • Giorgio Giorgi

    (University of Pavia)

  • Bienvenido Jiménez

    (Universidad Nacional de Educación a Distancia (UNED))

  • Vicente Novo

    (Universidad Nacional de Educación a Distancia (UNED))

Abstract

We extend the so-called approximate Karush–Kuhn–Tucker condition from a scalar optimization problem with equality and inequality constraints to a multiobjective optimization problem. We prove that this condition is necessary for a point to be a local weak efficient solution without any constraint qualification, and is also sufficient under convexity assumptions. We also state that an enhanced Fritz John-type condition is also necessary for local weak efficiency, and under the additional quasi-normality constraint qualification becomes an enhanced Karush–Kuhn–Tucker condition. Finally, we study some relations between these concepts and the notion of bounded approximate Karush–Kuhn–Tucker condition, which is introduced in this paper.

Suggested Citation

  • Giorgio Giorgi & Bienvenido Jiménez & Vicente Novo, 2016. "Approximate Karush–Kuhn–Tucker Condition in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 70-89, October.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:1:d:10.1007_s10957-016-0986-y
    DOI: 10.1007/s10957-016-0986-y
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    References listed on IDEAS

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    1. R. Andreani & J. M. Martinez & M. L. Schuverdt, 2005. "On the Relation between Constant Positive Linear Dependence Condition and Quasinormality Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 473-483, May.
    2. Joydeep Dutta & Kalyanmoy Deb & Rupesh Tulshyan & Ramnik Arora, 2013. "Approximate KKT points and a proximity measure for termination," Journal of Global Optimization, Springer, vol. 56(4), pages 1463-1499, August.
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    5. Gabriel Haeser & María Laura Schuverdt, 2011. "On Approximate KKT Condition and its Extension to Continuous Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(3), pages 528-539, June.
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    Cited by:

    1. Gabriele Eichfelder & Leo Warnow, 2021. "Proximity measures based on KKT points for constrained multi-objective optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 63-86, May.
    2. 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.
    3. Gabriel Haeser & Alberto Ramos, 2020. "Constraint Qualifications for Karush–Kuhn–Tucker Conditions in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 469-487, November.
    4. Marius Durea & Radu Strugariu, 2020. "On the sensitivity of Pareto efficiency in set-valued optimization problems," Journal of Global Optimization, Springer, vol. 78(3), pages 581-596, November.
    5. C. Gutiérrez & L. Huerga & B. Jiménez & V. Novo, 2020. "Optimality conditions for approximate proper solutions in multiobjective optimization with polyhedral cones," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 526-544, July.
    6. Roberto Andreani & Ellen H. Fukuda & Gabriel Haeser & Daiana O. Santos & Leonardo D. Secchin, 2024. "Optimality Conditions for Nonlinear Second-Order Cone Programming and Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 1-33, January.
    7. P. Kesarwani & P. K. Shukla & J. Dutta & K. Deb, 2022. "Approximations for Pareto and Proper Pareto solutions and their KKT conditions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 123-148, August.
    8. Giorgio Giorgi, 2018. "A Guided Tour in Constraint Qualifications for Nonlinear Programming under Differentiability Assumptions," DEM Working Papers Series 160, University of Pavia, Department of Economics and Management.
    9. Min Feng & Shengjie Li, 2018. "An approximate strong KKT condition for multiobjective optimization," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 489-509, October.

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