3 Location of critical failure surface, convergence and other problems
3.7 Sensitivity of the global optimization parameters on the performance of the global optimization method
In all of the heuristic global optimization methods, there is no simple rule to determine the parameters used in the analysis. In general, these parameters are established by experience and numerical tests. It is surprising to find that the sensitivity of different global optimization methods with respect to different parameters is seldom considered in the past, and the sensitivity of the parameters in slope stability analysis has not been reported. The authors consider this issue to be important for geotechnical engineering problems as there are different topographies, sub-soil conditions, ground water condi-tions, soil parameters, soil nails and external loads controlling the problem.
It appears that many researchers have not appreciated the importance of the parameters used for the global optimization. The sensitivity of each param-eter can be obtained through the nine numerical tests by the statistical orthogonal tests given in Tables 3.11–3.17 for examples 1 and 3. If the F value (Factorial Analysis of Variance after Fisher; Fisher and Yates, 1963) of one parameter is larger than the critical value F0.05 and is smaller than F0.01, it implies that the calculated result is sensitive to this parameter; otherwise if the F value is smaller than F0.05, the result is insensitive to this parameter. If the F value is larger than F0.01, the result is hyper-sensitive to this parameter.
For the simple problem given by example 1, every method can be worked with satisfaction for different optimization parameters. For example 3 which is a dif-ficult problem with a soft band (similar to a Dirac function), the Tabu search and the ant-colony methods are poor in performance (the domain transformation technique is not used) while all the other optimization methods are basically Table 3.11 The effects of parameters on SA analysis for examples 1 and 3 (Fλ= 0.14,
Ft0= 0.47, FN= 3.86, F0.05= 7.7, F0.01= 21.2)
1 – 0.5 1 – 10.0 1 – 100 Ex. 1 Ex. 3 NOFs
2 – 0.8 2 – 20.0 2 – 300
Ex. 1 Ex. 3
1 1 1 1 1.7256 1.3232 176,562 140,522
2 2 2 2 1.7241 1.2990 915,902 956,523
3 1 2 2 1.7235 1.2514 339,602 408,482
4 2 1 1 1.7264 1.2745 423,522 349,422
5 2 1 2 1.7258 1.2846 852,602 986,534
6 1 2 1 1.7239 1.3193 183,422 135,262
7 2 2 1 1.7262 1.3213 463,782 360,242
8 1 1 2 1.7268 1.2582 492,122 252,302
acceptable with the F value less than F0.05. The efficiency of different methods for this case is however strongly related to the choice of the parameters, unless the transformation technique by Cheng (2007) is adopted, which is equivalent to the use of a random number with more weighting in the soft-band region.
Every global optimization method can be tuned to work well if suitable opti-mization parameters or an initial trial are adopted. Since the suitable optiopti-mization Table 3.12 The effects of parameters on GA analysis for examples 1 and 3 (Fρc=
0.18, Fρm= 0.38, F0.05= 161.4, F0.01= 4052)
ρc ρm Results NOFs
1 – 0.85 1 – 0.001 Ex. 1 Ex. 3 Ex. 1 Ex. 3
2 – 0.95 2 – 0.1
1 1 1 1.7273 1.2849 80,384 94,418
2 1 2 1.7266 1.2794 52,544 40,626
3 2 1 1.7272 1.2767 89,806 104,116
4 2 2 1.7266 1.2998 58,612 45,224
Table 3.13 The effects of parameters on PSO analysis for examples 1 and 3 (Fc1= 0.60, Fc2= 0.37, Fω= 0.52, F0.05= 7.7, F0.01= 21.2)
c1 c2 ω Results NOFs
1 – 1.0 1 – 1.0 1 – 0.3 Ex. 1 Ex. 3 Ex. 1 Ex. 3 2 – 3.0 2 – 3.0 2 – 0.8
1 1 1 1 1.7287 1.4430 59,200 45,200
2 2 2 2 1.7401 1.2671 59,200 45,200
3 1 2 2 1.7353 1.2692 59,200 94,800
4 2 1 1 1.7226 1.2368 108,400 231,800
5 2 1 2 1.7309 1.2545 59,800 46,400
6 1 2 1 1.7269 1.2405 75,600 57,600
7 2 2 1 1.7376 1.2747 59,200 45,200
8 1 1 2 1.7266 1.2479 59,200 58,200
Table 3.14 The effects of parameters on SHM analysis for examples 1 and 3 (FHR= 1.91, FPR= 1.13, F0.05= 161.4, F0.01= 4052)
HR PR Results NOFs
1 – 0.80 1 – 0.05 Ex. 1 Ex. 3 Ex. 1 Ex. 3
2 – 0.95 2 – 0.10
1 1 1 1.7330 1.2947 57,717 180,221
2 1 2 1.7438 1.3748 57,763 81,194
3 2 1 1.7231 1.2799 107,191 68,529
4 2 2 1.7259 1.2824 57,931 118,340
= 0.97, FPR= 0.07, FNhm= 0.10, Fδ= 0.26, F0.05= 10.1, F0.01= 34.1)
HR PR Nhm δ Results NOFs
1 – 0.80 1 – 0.05 1 – 0.1 1 – 0.3 Ex. 1 Ex. 3 Ex. 1 Ex. 3 2 – 0.95 2 – 0.10 2 – 0.3 2 – 0.8
1 1 1 1 1 1.7348 1.2838 7654 13,547
2 1 1 1 2 1.7323 1.3523 13,654 9545
3 1 2 2 1 1.7295 1.3102 31,446 34,436
4 1 2 2 2 1.7347 1.3025 31,446 45,235
5 2 1 2 1 1.7270 1.2976 17,159 27,053
6 2 1 2 2 1.7271 1.2874 16,986 26,591
7 2 2 1 1 1.7273 1.2989 9640 15,219
8 2 2 1 2 1.7271 1.2878 19,621 13,128
Table 3.16 The effects of parameters on Tabu analysis for examples 1 and 3 (Fd= 0.63, FNt= 0.17, FHR= 0.49, FPR= 1.43, F0.05= 10.1, F0.01= 34.1)
d Nt HR PR Results NOFs
1 – 2 1 – 30 1 – 0.80 1 – 0.05 Ex. 1 Ex. 3 Ex. 1 Ex. 3 2 – 5 2 – 50 2 – 0.95 2 – 0.10
1 1 1 1 1 1.7411 1.5714 58,388 44,168
2 1 1 1 2 1.7413 1.5391 58,188 44,168
3 1 2 2 1 1.7424 1.5661 58,188 44,168
4 1 2 2 2 1.7413 1.5661 58,188 44,168
5 2 1 2 1 1.7429 1.5661 58,588 44,168
6 2 1 2 2 1.7415 1.5427 58,188 44,168
7 2 2 1 1 1.7415 1.5470 58,188 44,368
8 2 2 1 2 1.7354 1.5561 59,188 44,168
Table 3.17 The effects of parameters on ant-colony analysis for examples 1 and 3 (Fμ= 11.8, FQ= 0.002, Fd= 39.7, F0.05= 7.7, F0.01= 21.2)
μ Q d Results NOFs
1 – 0.3 1 – 10.0 1 – 10 Ex. 1 Ex. 3 Ex. 1 Ex. 3 2 – 0.8 2 – 50.0 2 – 20
1 1 1 1 1.7447 1.5332 83,500 76,200
2 2 2 2 1.7404 1.9239 66,800 50,800
3 1 2 2 1.7636 1.8787 66,800 50,800
4 2 1 1 1.7717 1.7420 83,500 76,200
5 2 1 2 1.7377 1.9239 66,800 50,800
6 1 2 1 1.7538 1.5049 83,500 76,200
7 2 2 1 1.7569 1.7420 83,500 76,200
8 1 1 2 1.7591 1.8435 66,800 50,800
parameters or the initial trial is difficult to be established for a general problem, the performance of a good optimization method should be relatively insensitive to these factors. Based on the numerical examples and the two special cases shown in Figures 3.26 and 3.27 and some other internal studies by the authors, the general comments on the different heuristic artificial intelligence-based global optimization methods are:
1 For normal and simple problems, practically every method can work well.
The harmony method and the genetic algorithm are the most efficient methods when the number of control variables is less than 20. The Tabu search and the ant-colony method are sometimes extremely efficient in the optimization process, but the efficiency of these two methods fluctuate sig-nificantly between different problems and are not recommended.
2 For normal and simple problems where the number of control variables exceeds 20, the MHM and the PSO are the recommended solutions as they are more efficient in the solution, and the solution time will not vary significantly between different problems.
3 For more complicated problems or when the number of control vari-ables is great, the effectiveness and efficiency of the PSO is nearly the best in all of the examples.
4 A thin soft band creates great difficulty in the global optimization analysis and the PSO will be the best method in this case. However, using the domain transformation strategy by Cheng (2007), all the global optimization methods can work well for this case.
5 For problems where an appreciable amount of trial failure surfaces will fail to converge, the simulated annealing method and the PSO are the recommended solutions.
In view of the differences in the performance between different global optimization algorithms, a more satisfactory solution is the combined use of two different algorithms. For example, the PSO or the MHM can be adopted for normal problems, while the SA can be adopted when the ‘failure to convergence’ counter is high. Further improvement can be achieved by using the optimized results from a particular optimization method as a good initial trial, and a second optimization method adopts the optimized result from the first optimization algorithm for the second stage of optimization with a reduced solution domain for each control variable.