Positivity

, Volume 17, Issue 3, pp 711–732 | Cite as

Nonsmooth multiobjective programming: strong Kuhn–Tucker conditions

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Abstract

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given in such a way that they generalize the classical ones, when the functions are differentiable. The relationships between them are analyzed. Then, we establish strong Kuhn–Tucker necessary optimality conditions in terms of the Clarke subdifferentials such that the multipliers of the objective function are all positive. Furthermore, sufficient optimality conditions under generalized convexity assumptions are derived. Moreover, the concept of efficiency is used to formulate duality for nonsmooth multiobjective problems. Wolf and Mond–Weir type dual problems are formulated. We also establish the weak and strong duality theorems.

Keywords

Multiobjective programming Optimality conditions Nonsmooth optimization Duality Constraint qualification 

Mathematics Subject Classification (2000)

90C46 90C29 49J52 
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Notes

Acknowledgments

The second author was partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran.

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FasaFasaIran
  2. 2.Department of MathematicsUniversity of IsfahanIsfahanIran

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