Self-Adaptive Genetic Algorithms with Simulated Binary Crossover

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SBX performance 1e+10 1 1e-10 Best function value Standard deviation in 15-th var 1e-20 0 500 1000 1500 2000 2500 Generation number 3000 3500 4000 Figure 13: Population-best objective function value and the standard deviation ofx15variable are shown for real-parameter GAs with SBX operator on function F2-2. Population best function value 1 1e-10 SBX Self-Adaptive ES 1e-20 0 200 400 600 Generation number 800 1000 Figure 15: Population-best objective function value for real-parameter GAs and self-adaptive ESs are shown for the elliptic function F2-3. Self-adaptive (15/15,100)-ES Performance 1e+10 1 1e-10 Best function value Mutation strength in 15-th var 1e-20 0 500 1000 1500 2000 2500 Generation number 3000 3500 4000 Figure 14: Population-best objective function value and the mutation strength forx15variable are shown for self-adaptive (15/15,100)-ES on function F2-2. Population best function value 1 1e-10 Self-Adaptive ES SBX 1e-20 0 200 400 600 Generation number 800 1000 Figure 16: Population-best objective function value with real-parameter GAs and self-adaptive ESs are shown for the multi-modal function F2-4. Clearly, a better crossover operator handling the linkage issue but with the concept of probability distribution to create children solutions is in order to solve such problems faster. One such implementation is suggested in Section 6. Besides the linkage issue discussed above, there is another mismatch between the GAs with SBX and the correlated self-adaptive ESs used above. In GAs, a selection pressure of 3 (best solution in a population gets a maximum of three copies after the tournament selection operation), whereas in the correlated selfadaptive (4,100)-ESs, only 4 best solutions are picked from 100 children solutions. In order to alleviate this mismatch in selection pressures, we use a different algorithm (we call ES-SBX) where a(;)-ES is used, but children solutions are created from parent solutions only by the action of SBX operator alone. Each parent (x(1;t)) mates with other parents in exactly=crossovers, everytime creating one child solution using equation 5. Like the SBX operator used in GAs, every variable is crossed with a probability 0.5. If a variable is not to be crossed, its value (x(i1;t)) in the first parent is directly passed on to the child. No explicit mutation operator is used. The rest of the algorithm is exactly the same as that in a(;)-ES. As shown in Figure 17, this new algorithm with (4,100)-ES-SBX is able to achieve 16
Population-best objective function value 100 10 1 0.1 0.01 0.001 0.0001 1e-05 1e-06 Correlated ES Non-isotropic ES SBX ES-SBX 1e-07 0 200 400 600 800 1000 Generation number Figure 17: The best objective function value in the population is shown for four evolutionary algorithms— real-parameter GAs with SBX, non-isotropic (4,100)-ES, and correlated self-adaptive (4,100)-ES, and (4,100)-ES-SBX—on function F3-1. MaximizevTx?dk(vTx)v?xk; much better performance than GAs with SBX operator, although no explicit pair-wise correlations among variables are used in any operator. However, we recognize that the linkage issue may still be a problem in general, but this simple change in the algorithm seems to fair well in a problem where finding pair-wise linkage information is important. 5.4 Ridge model Above problems have tested the ability of algorithms to converge near the true optimum with increasing precision. We now test algorithms for an opposing characteristic. We consider the following function, which is largely known as the ridge functions in ES literature: (24) wherexis theN-dimensional variable vector andvis the ridge axis (or the direction vector specifying the ridge axis). Thus, the first term is the projection ofxvector along the ridge axis. The term insidekkis the orthogonal component specifying the distance to the ridge axis. Since the objective is to maximize the overall function, the subgoals are to maximize the distance along the ridge axis and minimize the distance to the ridge axis. The parametersdand govern the shape of ridge functions. Higher values ofdmakes the second term more prominent compared to the first term and therefore makes an algorithm difficult to progress along the ridge axis. In all simulations here, we use a comparatively larged=1. The parameter has a direct effect on the fitness landscape. Usually, two values of are commonly used—=2is known as the parabolic ridge function and=1is known as the sharp ridge function. Here, we shall consider the parabolic ridge function only. The ridge functions have their theoretical maximum solution at infinity along the ridge axis in the search space. Since the maximum solution lies on the ridge axis, this function tests two aspects: converging ability on the axis, and diverging ability in the direction of the ridge axis. We test the performance of realparameter GAs with SBX and the non-isotropic self-adaptive ES. 17
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: Population standard deviation in x
Page 19 and 20: tical~vis used for all algorithms.
Page 21 and 22: help us to create problem specific
Page 23 and 24: Deb, K. and Agrawal, R. B. (1995) S

Self-Adaptive Genetic Algorithms with Simulated Binary Crossover

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