Crossover controlling the number of crossed genes (CCG) significantly improves the search performance of NSGA-II in many-objective optimization problems (MaOPs). Since the conventional NSGA-II deteriorates convergence of solutions toward the optimal Pareto front in MaOPs, to improve the search performance it is desirable to combine CCG crossover with a parent selection method based on fine-grained ranking of solutions. However, the search performance is not improved when CCG crossover is employed in IBEA_ε+, which also realizes fine-grained ranking of solutions. That is, in some cases, the search performance is not improved by CCG crossover depending on the parent selection method combined. To further improve the search performance of MOEAs in MaOPs, in this paper we propose a MOEA which employs CCG crossover with a particular parent selection based on fine-grained ranking of solutions. First, to clarify the feature of parent selection improving its search performance by employing CCG, we analyze difference of solutions search between NSGA-II and IBEA_ε+. As a result, we show that diversity preservation mechanism is absolutely imperative in parent selection to search Pareto optimal solutions of MaOPs broadly distributed in objective/variable space by using CCG crossover. That is, to further improve the search performance, it is necessity to make fine-grained rank of solutions while keeping diversity of solutions in parent selection. To satisfy this requirements in the parent selection, in this work we combine CCG crossover with self-controlling dominance area of solutions (S-CDAS) as the particular parent selection method. Through performance verification using many-objective knapsack problems with 4~10 objectives, we show that the search performance of S-CDAS combined with CCG crossover significantly improves by keeping the number of crossed genes very small. Also, we show that the effectiveness of CCG operator becomes significant as we increase the number of objectives and search performance of S-CDAS with CCG is higher than NSGA-II and IBEA_ε+ with CCG.

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The natural evolution strategies (NESs) is a family of iterative methods for black-box function optimization. Instead of directly minimizing an objective function, NESs minimizes the expectation of the objective function value over an arbitrary parametric probability distribution. In each iteration, NESs updates parameters of the distribution by using an estimated natural gradient of the expectation of the objective function value. Exponential NES (xNES) is an effective method of NESs that uses the multivariate normal distribution as the probability distribution. Since the shape of a normal distribution can take the form of a rotated ellipse in the solution space, xNES shows relatively good performance for ill-conditioned and non-separable objective functions. However, we believe that xNES has two problems that cause performance degradation. The first problem is that the spread of normal distribution tends to shrink excessively even if the distribution does not cover a (local) optimal point. This will cause premature convergence. The second problem is that the learning rates for the parameters of distribution are not appropriate. The learning rates depend only on the dimension of objective function although they should be designed depending on all the factors that influence the precision of natural gradient estimation. Moreover, they are set to small values for preventing the premature convergence and these results in too slow convergence speed even if the distribution covers the optimal point. In order to remedy the problems of xNES, we propose a new method of NESs named the distance-weighted exponential natural evolution strategy (DX-NES). On several benchmark functions, we confirmed that DX-NES outperforms xNES and that DX-NES shows better performance than CMA-ES on the almost all functions.

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