Improved step size adaptation for the MO-CMA-ES

The multi-objective covariance matrix adaptation evolution strategy (MO-CMA-ES) is an evolutionary algorithm for continuous vector-valued optimization. It combines indicator-based selection based on the contributing hypervolume with the efficient strategy parameter adaptation of the elitist covariance matrix adaptation evolution strategy (CMA-ES). Step sizes (i.e., mutation strengths) are adapted on individual-level using an improved implementation of the 1/5-th success rule. In the original MO-CMA-ES, a mutation is regarded as successful if the offspring ranks better than its parent in the elitist, rank-based selection procedure. In contrast, we propose to regard a mutation as successful if the offspring is selected into the next parental population. This criterion is easier to implement and reduces the computational complexity of the MO-CMA-ES, in particular of its steady-state variant. The new step size adaptation improves the performance of the MO-CMA-ES as shown empirically using a large set of benchmark functions. The new update scheme in general leads to larger step sizes and thereby counteracts premature convergence. The experiments comprise the first evaluation of the MO-CMA-ES for problems with more than two objectives.