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.
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Acknowledgments
This research was partially supported (for the second and third author) by the Ministerio de Economía y Competitividad (Spain) under project MTM2012-30942. The authors are grateful to the associate editor and the anonymous referees for their helpful comments and suggestions.
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Communicated by Fabian Flores-Bazán.
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Giorgi, G., Jiménez, B. & Novo, V. Approximate Karush–Kuhn–Tucker Condition in Multiobjective Optimization. J Optim Theory Appl 171, 70–89 (2016). https://doi.org/10.1007/s10957-016-0986-y
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DOI: https://doi.org/10.1007/s10957-016-0986-y
Keywords
- Approximate optimality conditions
- Sequential optimality conditions
- Enhanced Fritz John conditions
- Enhanced Karush–Kuhn–Tucker conditions
- Vector optimization problems