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On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization

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  • Regina S. Burachik

    (University of South Australia)

  • M. M. Rizvi

    (University of South Australia)

Abstract

We consider a smooth multiobjective optimization problem with inequality constraints. Weak Kuhn–Tucker (WKT) optimality conditions are said to hold for such problems when not all the multipliers of the objective functions are zero, while strong Kuhn–Tucker (SKT) conditions are said to hold when all the multipliers of the objective functions are positive. We introduce a new regularity condition under which (WKT) hold. Moreover, we prove that for another new regularity condition (SKT) hold at every Geoffrion-properly efficient point. We show with an example that the assumption on proper efficiency cannot be relaxed. Finally, we prove that Geoffrion-proper efficiency is not needed when the constraint set is polyhedral and the objective functions are linear.

Suggested Citation

  • Regina S. Burachik & M. M. Rizvi, 2012. "On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 477-491, November.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:2:d:10.1007_s10957-012-0078-6
    DOI: 10.1007/s10957-012-0078-6
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    References listed on IDEAS

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    1. White, D. J., 1983. "Concepts of proper efficiency," European Journal of Operational Research, Elsevier, vol. 13(2), pages 180-188, June.
    2. V. Preda & I. Chiţescu, 1999. "On Constraint Qualification in Multiobjective Optimization Problems: Semidifferentiable Case," Journal of Optimization Theory and Applications, Springer, vol. 100(2), pages 417-433, February.
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    4. G. Giorgi & B. Jiménez & V. Novo, 2009. "Strong Kuhn–Tucker conditions and constraint qualifications in locally Lipschitz multiobjective optimization problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(2), pages 288-304, December.
    5. M. C. Maciel & S. A. Santos & G. N. Sottosanto, 2009. "Regularity Conditions in Differentiable Vector Optimization Revisited," Journal of Optimization Theory and Applications, Springer, vol. 142(2), pages 385-398, August.
    6. X. F. Li & J. Z. Zhang, 2005. "Stronger Kuhn-Tucker Type Conditions in Nonsmooth Multiobjective Optimization: Locally Lipschitz Case," Journal of Optimization Theory and Applications, Springer, vol. 127(2), pages 367-388, November.
    7. G. Bigi & M. Pappalardo, 1999. "Regularity Conditions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 102(1), pages 83-96, July.
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    Cited by:

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    5. Manxue You & Shengjie Li, 2017. "Separation Functions and Optimality Conditions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 527-544, November.

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