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This paper presents a new evolutionary algorithm for function optimization named the distance-weighted exponential natural evolution strategies (DX-NES). DX-NES remedies two problems of a conventional method, the exponential natural evolution strategies (xNES), that shows good performance when it does not need to move the distribution for sampling individuals down the slope to the optimal point. The first problem of xNES is that the search efficiency deteriorates while the distribution moves down the slope of an ill-scaled function because it degenerates before reaching the optimal point. The second problem is that the settings of learning rates are inappropriate because they do not taking account of some factors affecting the estimate accuracy of the natural gradient. We compared the performance of DX-NES with that of xNES and CMA-ES on typical benchmark functions and confirmed that DX-NES outperformed the xNES on all the benchmark functions and that DX-NES showed better performance than CMA-ES on the almost all functions except the k-tablet function.