How to Maintain at Most N Well-Distributed, Non-dominated Solutions to a Pareto Optimization Problem

We consider the question: given a multi-objective optimization problem and a search algorithm for generating solutions to it, which non-dominated solutions should be kept, assuming at most N of them can be stored at any time? One would like the points (the images of the solutions in the objective space) stored to form a well-distributed approximation of the Pareto front of the sequence of points generated by the search algorithm – a notion that can be formally captured by the concepts of e-dominance and e-approximate sets. We focus on three ‘archiving’ algorithms designed specifically for storing a size-bounded approximation set of a sequence of non-dominated solutions, as they are generated: two are based directly on e-dominance and the other employs an adaptively changing grid in the objective space. We consider how these three archiving schemes perform when one can store at most N points to approximate a sequence. We show that certain types of objective space (or sequences) can cause the e-dominance based approaches to discard ‘too many’ points so that much fewer than N are obtained, giving a ‘lower resolution’ approximation than is obtained by the adaptive grid method.