Many-criteria dominance relations

Abstract

Next to the Pareto dominance relation, alternative order relations can be useful in many-objective optimization. In particular, it is interesting to extend the Pareto dominance relation in order to make more pairs comparable and decrease the size of the optimal set (for discrete approximations in continuous optimization or discrete problems), which tends to grow exponentially with an increasing number of objectives. Here, we review some basic concepts in order theory with a focus on the concept of an order extension. Moreover, we will define and discuss properties of some orders that have been proposed as alternatives to the Pareto dominance relation in the context of dealing with larger number of objective functions. The chapter compares the different concepts and reveals commonalities between them. It will be shown that many of the order extensions proposed are special cases of cone orders. Therefore, the chapter puts particular emphasize on the concept of dominance cones and demonstrates how different ways of defining dominance cones, such as by means of angles or by trade-offs, can be mapped onto each other.