Fundamentals of Optimization by Mark French

Author:Mark French
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

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Figure 4.40 shows this function in 3D. The minimum occurs at x = −1.238, y = −1.238.

Fig. 4.40Test function for evolutionary algorithm example

Figure 4.41 shows a contour plot of the test function. It’s clear that there are starting points that would require a circuitous route to the minimum.

Fig. 4.41Contour plot of evolution algorithm test function

If, for example, the starting point were x = 0, y = −4, the search direction for the steepest descent method would point very nearly at the minimum. If, in contrast, the starting point was x = 1, y = 2, the search direction would point well away from the minimum. Indeed, the search path would need to arc around the peak of the objective function at x = 0.369, y = 0.369.

Figure 4.42 shows the first iteration or generation perhaps. The box shows the starting point, and the circles show the randomly chosen points where objective function values are calculated. Finally, the diamond shows the minimum – the point the algorithm is trying to find. There are ten random points calculated for each iteration. There needs to be some range defined for x and y. For this example, both were allowed to vary by ± 2 from the starting point. The minimum point is the one at the upper left.

Fig. 4.42Iteration 1, starting point, x = 1, y = 2, f = 0.717

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