Slopes, Error Bounds and Weak Sharp Pareto Minima of a Vector-Valued Map


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.

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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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Correspondence to Xuan Duc Ha Truong.

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Communicated by Marcin Studniarski.

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Truong, X.D.H. Slopes, Error Bounds and Weak Sharp Pareto Minima of a Vector-Valued Map. J Optim Theory Appl 176, 634–649 (2018).

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  • Vector-valued map
  • Slope
  • Error bound
  • Weak sharp Pareto minima

Mathematics Subject Classification

  • 49J53
  • 58C06
  • 90C29