Abstract
Analyzing the behavior and stability properties of a local optimum in an optimization problem, when small perturbations are added to the objective functions, are important considerations in optimization. The tilt stability of a local minimum in a scalar optimization problem is a well-studied concept in optimization which is a version of the Lipschitzian stability condition for a local minimum. In this paper, we define a new concept of stability pertinent to the study of multiobjective optimization problems. We prove that our new concept of stability is equivalent to tilt stability when scalar optimizations are available. We then use our new notions of stability to establish new necessary and sufficient conditions on when strict locally efficient solutions of a multiobjective optimization problem will have small changes when correspondingly small perturbations are added to the objective functions.
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Acknowledgements
The authors are indebted to Prof. Boris Mordukhovich for his serious discussions on the earlier draft of this manuscript to enrich this manuscript. Moreover, the authors are grateful for the editor and the two anonymous referees for their valuable and constructive comments to improve this manuscript. They would also like to extend their thankfulness to Maxie Schmidt and Karim Rezaei for the linguistic editing of the final draft of this paper.
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Communicated by Fabián Flores-Bazán.
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Sadeghi, S., Mirdehghan, S.M. Stability of Local Efficiency in Multiobjective Optimization. J Optim Theory Appl 178, 591–613 (2018). https://doi.org/10.1007/s10957-018-1312-7
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DOI: https://doi.org/10.1007/s10957-018-1312-7