A parameterless performance metric for reference-point based multi-objective evolutionary algorithms

Most preference-based multi-objective evolutionary algorithms use reference points to articulate the decision maker's preferences. Since these algorithms typically converge to a sub-region of the Pareto-optimal front, the use of conventional performance measures (such as hypervolume and inverted generational distance) may lead to misleading results. Therefore, experimental studies in preference-based optimization often resort to using graphical methods to compare various algorithms. Though a few ad-hoc measures have been proposed in the literature, they either fail to generalize or involve parameters that are non-intuitive for a decision maker. In this paper, we propose a performance metric that is simple to implement, inexpensive to compute, and most importantly, does not involve any parameters. The so called expanding hypercube metric has been designed to extend the concepts of convergence and diversity to preference optimization. We demonstrate its effectiveness through constructed preference solution sets in two and three objectives. The proposed metric is then used to compare two popular reference-point based evolutionary algorithms on benchmark optimization problems up to 20 objectives.