Self-Adaptive Genetic Algorithms with Simulated Binary Crossover

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useN=30 5.4.1 Function F4-1: Parabolic ridge Here we consider the ridge function with=2. At first, we choosev1=1and all othervi=0for i=2;:::;N. This makes the first coordinate axis as the ridge axis. In all simulations, we and the population is initialized inxi2[?2;2]. Real-parameter GAs with SBX (=1) and with binary tournament selection are used. A population of size 100 is used. Figure 18 plots the distance along the ridge axis (the termvTx) with generation number. The figure shows that real-parameter GAs are able to 1e+120 1e+100 ES-SBX ES 1e+80 1e+60 1e+40 SBX 1e+20 BLX-0.5 1 1e-20 0 50 100 150 200 250 300 350 400 450 500 Generation Number Figure 18: Performance of real-parameter GAs with SBX and BLX-0.5, non-isotropic selfadaptive (10/10,100)-ESs, and (10,100)-ES-SBX inx1 1e+100 1e+90 1e+80 1e+70 1e+60 1e+50 1e+40 ES, mut str in x_1 SBX, std dev in x_1 1e+30 ES, mut str in x_2 1e+20 1e+10 SBX, std dev in x_2 1 1e-10 0 50 100 150 200 250 300 350 400 450 500 Generation number Figure 19: Population standard deviation andx2for GAs with SBX and population average mutation strengths inx1andx2are shown for are shown on the ridge function F4-1. the ridge function F4-1. Distance along ridge axis progress towards infinity along the ridge axis exponentially with generation number (the ordinate axis is plotted in logarithmic scale). A run with non-isotropic self-adaptive (10/10,100)-ES with one independent mutation strength parameter for each variable shows a similar behavior, but with much better progress rate. Since the optimum is at infinity along the ridge axis, an algorithm with faster divergence characteristics is desired. Since BLX-0.5 allows creation of more diverse children solutions than parents, we tried using real-paramater GAs with BLX-0.5 operator. The figure shows poor self-adaptive power of the BLX-0.5 operator. Figure 19 shows the population standard deviation inx1andx2for real-parameter GAs with SBX operator. The figure also shows the population average mutation strengths inx1andx2as they evolve for the parabolic ridge function. It is clear that in both algorithms the population diversity or mutation strength forx1grows much faster than that inx2. This is what we have expected, because for the parabolic ridge function there are two subgoals. By increasing the population diversity or mutation strength at faster rate will allow the corresponding algorithm to move quicker along the ridge axis. On the other hand, since the secondary goal is to come closer to the ridge axis, an ideal situation would be reduce the population diversity or mutation strengths of other variables (x2toxN) small as possible. However, both algorithms resorted to increase these quantities with generation, but at a rate much smaller than the growth inx1variable. Population diversity or mutation strength as Next, we apply (10,100)-ES-SBX (where SBX is used as the only search operator in an ES, which is described in the previous subsection). SBX operator with=1is used. Figure 18 shows that a better progress compared to all other algorithms is obtained with ES-SBX algorithm. 5.4.2 Function F4-2: Parabolic ridge with rotated ridge axis In this case, we use a random~v, so that the ridge axis is now not along a coordinate direction. Parameters as that used in F4-1 for real-parameter GAs with SBX and self-adaptive ES are chosen here and an iden- 18
tical~vis used for all algorithms. Figure 20 shows the progress towards the ridge axis with real-parameter GAs, with non-isotropic self-adaptive ESs, and with ES-SBX algorithm. The self-adaptive ES has a faster progress rate. However, it is worth mentioning that a non-recombinative self-adaptive (10,100)-ES performs poorly in this function. However, notice that due to the parameter interactions, the progress along the Distance along ridge axis 200 180 160 140 120 100 80 60 40 20 0 SBX -20 0 500 1000 Generation Number 1500 2000 ES ES-SBX BLX-0.5 Figure 20: Performance of real-parameter GAs with SBX and BLX-0.5, non-siotropic selfadaptive (10/10,100)-ESs, and (10,100)-ES-SBX are shown on the rotated ridge function F4-2. inx1 100 SBX, std dev in x_1 SBX, std dev in x_2 ES, mut str in x_1 10 ES, mut str in x_2 1 0.1 0.01 0.001 0 500 1000 1500 2000 Population diversity or mutation strength Figure 21: Population standard deviation Population diversity or mutation strength andx2for GAs with SBX and population average mutation strengths inx1andx2are shown for the rotated ridge function F4-2. ridge axis is now not exponential to the generation number, rather the progress is linear. Since to improve along the ridge, allNvariables need to be changed in a particular way, the progress slows down. However, both algorithms have been able to maintain a steady progress. Real-parameter GAs with BLX-0.5 also has progress towards optimum, but the progress is slow. The (10,100)-ES-SBX algorithm (with=1) is able to find better performance than GAs with the SBX operator. Figure 21 shows the population average standard deviation inx1andx2for real-parameter GAs with SBX operator and the population average mutation strength inx1andx2. In contrary to their evolution in the parabolic ridge F4-1 (Figure 19), here, these quantities in both algorithms reduce and fluctuate in a certain range. More importantly, these quantities for variablesx0andx1now varies in the same range. This can also be explained as follows. In the rotated ridge function, the ridge axis~vis a random direction, other than any coordinate direction. For the same reason, any orthogonal direction to the ridge axis is also a non-coordinate direction. In order to simultaneously satisfy both subgoals of maximizing progress towards the ridge axis and minimizing the distance to the ridge axis, the best an algorithm can do is to have a compromise in the rate of growth in each variable direction. In this case, both algorithms make careful changes to variables by keeping the population diversity or the mutation strength small so that a compromise in both subgoals is achieved. Naturally, this reduces the overall progress rate along the ridge axis. However, it may be mentioned here that such a linear behavior of rotated ridge functions is inevitable for any self-adaptive strategies which work with adaptations in variable directions independently. For a strategy which has the capability to independently adapt variations in arbitrary directions (not necessarily the coordinate directions), an exponential progress towards optimum with generation number may be possible (Hansen and Ostermeier, 1998). 6 Future Studies This study suggests a number of extensions, which are outlined in the following: 1. Real-parameter GAs with SBX operator can be compared with other self-adaptive ES implementations. 19
Page 1 and 2: Self-Adaptive Genetic Algorithms wi
Page 3 and 4: (?0:5x(1;t) suggested. [x(1;t) i+1:
Page 5 and 6: Step 1: Choose a random numberu2[0;
Page 7 and 8: x(t+1) i=(t) i=x(t) (t+1) for(0) ie
Page 9 and 10: F1-1:MinimizeNXi=1(xi?xi)2; 5.1.1 F
Page 11 and 12: Population-best distance from optim
Page 13 and 14: that when a small initial mutation
Page 15 and 16: Population standard deviation in x
Page 17: Population-best objective function
Page 21 and 22: help us to create problem specific
Page 23 and 24: Deb, K. and Agrawal, R. B. (1995) S

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