Abstract

In many situations it is necessary to consider problems in which decisions have to be taken in a given sequence. For this purpose the lexicographic order of Rn\mathbb{R}^n appears to be an appropriate concept to employ. Therefore, we will study several of its properties in this paper. In a first step we introduce some definitions and regularity concepts with respect to this type of order. Secondly, we prove several classical results redesigned for the vectorial setting as well as several new specific results which are interesting consequences of our approach. Finally, as an application, we study min-max control problems with order. We investigate the existence of the value and we provide results concerning the existence of saddle points for a vectorial cost.

Author information

Contact details are reproduced from the original publication and may be historical.

Nathalie Caroff

Lab. de Mathématiques et Physique, Université de Perpignan, 66860 Perpignan, France

Oana Silvia Serea

Dept. of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, 5063 Bergen, Norway
and: Lab. de Mathématiques et Physique, Université de Perpignan, 66860 Perpignan, France

oss@hvl.no

Keywords

  • Lexicographic order
  • saddle points
  • vectorial games

Mathematics Subject Classification

26B25, 26B25, 49K35, 49K15, 90C47

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Suggested citation

N. Caroff, O. S. Serea. “Ky-Fan and Sion Theorems for the Lexicographic Order and Applications to Vectorial Games and Min-Max Control Problems.” Journal of Convex Analysis 27 (2020), No. 3, 1051–1072. https://doi.org/10.68381/jca27055

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