In this paper, linear multiplicative programs are approached with a branch-and-bound scheme and a detailed computational study is provided. Several underestimation functions are analyzed and various partitioning criteria are presented. A particular class of linear multiplicative programs, useful to solve some applicative bilevel problems, is considered from a theoretical point of view to emphasize an efficient solution method. Detailed results of the computational study are provided to point out the performances provided by using various underestimation functions and partitioning criteria, thus improving some of the results of the current literature.

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. By using tools of variational analysis and normal operator techniques, very recently some authors have 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.