Fig. 3

a–e Five key frames that illustrate how four candidate individuals with different colors converge to the global minimum at (0, 0) under the hybrid algorithm on the contour of Griewank function. a The initial positions of the four candidates. b The positions of the four candidates after the first convergence of gradient decent but without evolutionary operation. c The positions of the four candidates after the first evolutionary operation. d The positions of the four candidates after the second evolutionary operation. e The final positions of the four candidates. The bar graph in f shows the number of global minimums found by GD, GD with random initialization, and GD with selective substitution algorithms in 100 instances, respectively