Kriging-surrogate-based optimization considering expected hypervolume improvement in non-constrained many-objective test problems

This paper presents a comparison of the criteria for updating the Kriging surrogate models in surrogate-based non-constrained many-objective optimization: expected improvement (EI), expected hypervolume improvement (EHVI), and estimate (EST). EI has been conventionally used as the criterion considering the stochastic improvement of each objective function value individually, while EHVI has been proposed as the criterion considering the stochastic improvement of the front of non-dominated solutions in multi-objective optimization. EST is the value of each objective function estimated non-stochastically by the Kriging model without considering its uncertainties. Numerical tests were conducted in the DTLZ test function problems (up to 8 objectives). It empirically showed that, in the DTLZ1 problem, EHVI has greater advantage of convergence and diversity to the true Pareto-optimal over EST and EI as the number of objective functions increases. The present results also suggested the expectation that the Kriging-surrogate-based optimization using EHVI may overcome the direct optimization without using the Kriging models when the number of objective functions becomes more than 10. In the DTLZ2 problem, however, EHVI achieved slower convergence to the true Pareto-optimal front than EST and EI. It is due to the complexity of objective function space and the selection of additional sample points.