Generalized ideal points
Andrzej M. J. Skulimowski · AIP conference proceedings · 2019
This paper presents an extension of the ideal point concept, which is one of the most widespread ideas in multicriteria optimization. In the simplest case, the ideal point of a multicriteria optimization problem is an element of the criteria space with coordinates equal to the optima of each criterion function considered as a scalar objective. First, we will provide an overview of the properties of ideal points and their earlier generalizations. Then, we will present an extension of ideal points in the case where the dimension of the criteria space is higher than 2. Specifically, scalar optima used in the classical definition of ideal points are replaced by Pareto optima with respect to criteria subsets. It is assumed that the generating criteria subsets belong to a minimal covering of the set of all criteria. We will distinguish several classes of generalized ideal points (GIP) so defined. For example, proper GIPs are calculated with respect to the criteria subsets that form a partition of the criteria set. Regular GIPs are those which correspond to a covering with criteria subsets of the same cardinality and such that all intersections of these sets also contain the same number of criteria or are empty. We will also present and briefly discuss the notion of local GIPs as well as applications of the GIP concept to formulating criteria independence conditions and to consensus finding in multicriteria games.