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Ekeland variational principles involving set perturbations in vector equilibrium problems

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Abstract

On the basis of the notion of approximating family of cones and a generalized type of Gerstewitz’s/Tammer’s nonlinear scalarization functional, we establish variants of the Ekeland variational principle (for short, EVP) involving set perturbations for a type of approximate proper solutions in the sense of Henig of a vector equilibrium problem. Initially, these results are obtained for both an unconstrained and a constrained vector equilibrium problem, where the objective function takes values in a real locally convex Hausdorff topological linear space. After that, we consider special cases when the objective function takes values in a normed space and in a finite-dimensional vector space. For the finite-dimensional objective space with a polyhedral ordering cone, we give the explicit representation of variants of EVP depending on matrices, and in such a way, some selected applications for multiobjective optimization problems and vector variational inequality problems are also derived.

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Notes

  1. f is said to be (qH)-lower semicontinuous from above ((qH)-lsca) at \(x\in X\) if, for each \(r \in \mathbb {R}\) and \(x_m\rightarrow x\), from \(f( x_1 )+ rq \le _H 0\) and \(f(x_{m+1})+ t_m q \le _H f(x_m)\), \(t_m\ge 0\), for all \(m \in \mathbb {N}\), it follows that \(f(x) + rq \le _H 0\). It is firstly introduced by the authors in [34].

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Acknowledgements

The author is grateful to the anonymous referees for their valuable remarks and suggestions. This work was supported by the Domestic Master/PhD Scholarship Programme of Vingroup Innovation Foundation [Grant Number VINIF.2019.TS.19] and The National Foundation for Science and Technology development (NAFOSTED) under grant no. 101.01-2020.23.

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Correspondence to Le Phuoc Hai.

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Hai, L.P. Ekeland variational principles involving set perturbations in vector equilibrium problems. J Glob Optim 79, 733–756 (2021). https://doi.org/10.1007/s10898-020-00945-5

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