We present a surrogate-assisted multiobjective optimization algorithm. The aggregation of the objectives relies on the Uncrowded Hypervolume Improvement (UHVI) which is partly replaced by a linear-quadratic surrogate that is integrated into the CMA-ES algorithm. Surrogating the UHVI poses two challenges. First, the UHVI is a dynamic function, changing with the empirical Pareto set. Second, it is a composite function, defined differently for dominated and nondominated points. The presented algorithm is thought to be used with expensive functions of moderate dimension (up to about 50) with a quadratic surrogate which is updated based on its ranking ability. We report numerical experiments which include tests on the COCO benchmark. The algorithm shows in particular linear convergence on the double sphere function with a convergence rate that is 6-20 times faster than without surrogate assistance.
CCS CONCEPTS• Mathematics of computing → Stochastic control and optimization; • Computing methodologies → Modeling methodologies; Model verification and validation.
In this paper, we present a comparative benchmark of two implementations of CMA-ES, both with and without diagonal decoding. The benchmarked variants of CMA-ES with diagonal decoding adaptively split the update of the covariance matrix into an update with the original CMA-ES method and an update with the separable-CMA-ES method. Thus, the diagonal decoding should allow for improved performance on separable functions with minimal loss on nonseparable ones. To gain insight into how diagonal decoding impacts CMA-ES runs, an assessment of the performance gain or loss due to the use of diagonal decoding relative to the original CMA-ES, was performed on bbob problems using the COCO platform. We were also interested in variances that might emerge from the difference in the implementations. The data presented in this paper shows improved performance of the CMA-ES on separable functions when using diagonal decoding, without any apparent loss on nonseparable ones. In addition, a few performance variances were spotted in the weakly structured functions, which appeared uncorrelated with the use of diagonal decoding. However, they can be traced back to implementation differences, such as the stopping conditions that may result in different runs, as the data suggests.
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