Quantitative Performance Assessment of Multiobjective Optimizers: The Average Runtime Attainment Function

Numerical benchmarking of multiobjective optimization algorithms is an important task needed to understand and recommend algorithms. So far, two main approaches to assessing algorithm performance have been pursued: using set quality indicators, and the empirical attainment function and its higher-order moments as a generalization of empirical cumulative distributions of function values. Both approaches have their advantages but rely on the choice of a quality indicator and/or take into account only the location of the resulting solution sets and not when certain regions of the objective space are attained. In this paper, we propose the average runtime attainment function as a quantitative measure of the performance of a multiobjective algorithm. It estimates, for any point in the objective space, the expected runtime to find a solution that weakly dominates this point. After defining the average runtime attainment function and detailing the relation to the empirical attainment function, we illustrate how the average runtime attainment function plot displays algorithm performance and differences in performance for some algorithms that have been previously run on the biobjective bbob-biobj test suite of the COCO platform.