Abstract
The aim of this work is to study a notion of variational convergence for vector-valued functions. We show that it is suitable for obtaining existence and stability results in convex multiobjective optimization. We obtain various of properties of the variational convergence. We characterize it via the set convergence of epigraphs, coepigraphs, level sets, and some infima. We also characterize it by means of two metrics. We compare it with other notions of convergence for vector-valued functions from the literature and we show that it is more general than most of them. For obtaining the existence and stability results we employ an asymptotic method that has shown to be very useful in optimization theory. In this method we couple the variational convergence with notions of asymptotic analysis (asymptotic cones and functions).
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Notes
The term recession function is employed in the literature when dealing with convex functions.
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Acknowledgments
The author wishes to express his gratitude to the two anonymous referees for their valuable remarks and suggestions. This work was supported by Proyecto FONDECYT 1100919 through CONICYT-Chile.
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López, R. Variational convergence for vector-valued functions and its applications to convex multiobjective optimization. Math Meth Oper Res 78, 1–34 (2013). https://doi.org/10.1007/s00186-013-0430-0
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DOI: https://doi.org/10.1007/s00186-013-0430-0
Keywords
- Asymptotic cone and function
- Efficient and weakly efficient minimizer
- Generalized \(\varepsilon \)-quasi minimizer
- Multiobjective optimization
- Set convergence
- Variational convergence