A variational approach to weakly continuous relations in Banach spaces

Optimization and equilibrium problems have been extensively studied when the involved preference relations admit a representation by means of real-valued functions. Although these problems have been analyzed under very minimal assumptions on the representation function, this context could appear to be quite restrictive in some practical situations. Using tools of variational analysis and normal operator techniques, very recently some authors explored the properties of preference relations that do not necessarily admit a numerical representation. However, these contributions are limited to finite-dimensional settings. Our aim in the present work is to develop a new analysis of preference relations in topological spaces and to analyze, in Banach spaces, a suitable concept of a normal operator to the upper contour set. As an application of our theoretical developments, we analyze a particular preference equilibrium problem (of which a preference maximization problem is a particular case) by using a suitable quasi-variational inequality formulation: as an example, a preference allocation problem is also considered.