Slopes, Error Bounds and Weak Sharp Pareto Minima of a Vector-Valued Map
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In this paper, we provide a detailed study of the upper and lower slopes of a vector-valued map recently introduced by Bednarczuk and Kruger. We show that these slopes enjoy most properties of the strong slope of a scalar-valued function and can be explicitly computed or estimated in the convex, strictly differentiable, linear cases. As applications, we obtain error bounds for lower level sets (in particular, a Hoffman-type error bound for a system of linear inequalities in the infinite-dimensional space setting, existence of weak sharp Pareto minima) and sufficient conditions for Pareto minima.
KeywordsVector-valued map Slope Error bound Weak sharp Pareto minima
Mathematics Subject Classification49J53 58C06 90C29
The research was carried out during the author’s stays at the Vietnam Institute for Advanced Study in Mathematics and the Institute of Mathematics, University of Erlangen-Nuremberg, under the Georg Forster grant of the Alexander von Humboldt Foundation, and was partially supported by NAFOSTED, Grant 101.01-2017.20. The author thanks the editor and the referees for the helpful comments and suggestions, which allowed to improve the paper.
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