8.7 Examples

In the following we apply the algorithm from Section 8.4.3 to three examples, namely two popular test functions [77] and a TCAD example. The first problem is to maximize the function

$$f_1:\quad [0,10]^2\to\mathbb{R},\quad (x,y)\mapsto x \sin(4x) + \frac{11}{10} y \sin(2y),$$ (8.1)

which has several extrema (cf. Figure 8.1). Its global maximum value is $\approx 19.863$ at the point $(\approx 9.824,10)$.

The second problem is to minimize the so called six hump camel back function

$$f_2:\quad [-3,3]\times[-2,2] \to \mathbb{R},\quad (x,y)\mapsto \left(\frac{x^4}{3}-2.1x^2+4\right) + xy + 4y^2(y^2-1).$$ (8.2)

It is shown in Figure 8.2 and has a global minimum value of $\approx -1.0316$ at the two points $(-0.0898,0.7126)$ and $(0.0898,-0.7126)$, approximately.

Figure 8.1: The function $f_1$ defined in (8.1). The maximum is at approximately $(9.824,10)$.

Figure 8.2: The function $f_2$ defined in (8.2). The minima are at approximately $(-0.0898,0.7126)$ and $(0.0898,-0.7126)$.

Figure 8.3: Comparison of the progress of the three optimizers DONOPT, GENOPT, and SIMAN. The figure shows the mean relative error plotted depending on the number of evaluations.

We used the algorithm as implemented in the EGO optimizer (cf. Section 9.3.1). For solving both optimization problems the populations size was $50$ individuals, the maximum number of generations was $200$, the mutation probability $0.2$, the ratio of $\sigma$ to the length of the interval $0.1$, and the number of steady state individuals was $5$. This standard configuration determined the global extrema of both functions every time in ten consecutive runs to at least all digits given above.

The third and last example stems from a TCAD application. Here six parameters were extracted from the drain currents of the select transistor of a storage cell matching two transfer characteristics for two bulk voltages and two times 27 points. MINIMOS NT [16] was used as the device simulator. We used three optimizers, namely DONOPT, GENOPT, and SIMAN (cf. Section 9.3).

Figure 8.3 shows the progress of the three optimizers. The local optimizer DONOPT does not yield a good result, although its initial progress is fast. Furthermore, the evaluations of the genetic algorithm, GENOPT, are better parallelized on a cluster of workstations than those of simulated annealing, SIMAN, and thus in terms of wall clock time elapsed, the genetic algorithm is the fastest optimizer.