Online optimization algorithms 205
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y = f(x)
RCDS, N=500
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y = f(x)
RSimplex, N=500
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y = f(x)
N-M Simplex, N=500
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y = f(x)
GP, N=300
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Figure 7.14 The convergence histories over 500 function evaluations of RCDS (top
left), RSimplex (top right), and N-M simplex (bottom left) and over 300 evaluations
for GP (bottom right) for the Rosenbrock-4 function with three noise levels.
level (σ = 0.001), all algorithms converge toward the Pareto front, although
the NSGA-II front is narrow and incomplete. With higher noise (σ = 0.01
or 0.1), the NSGA-II method fails to converge to the Pareto front, while the
PSO and MG-GPO algorithms converge to the same front for the medium or
high noise levels.
The Figure 7.16 left plot shows the minimum values for the two objective
functions over the course of optimization for the three algorithms for the lownoise case (σ = 0.001). The convergence speed of MG-GPO is much faster
than PSO, which is in turn much faster than NSGA-II. The cause of the slow
convergence for NSGA-II is the low diversity of its new trial solutions. The
diversity can be measured by the distribution of the crowding distance, here
defined as the distance of a solution to its nearest neighbor in the parameter
space among all previous solutions. The right plot shows the distribution of
the crowding distances for the three algorithms. Interestingly, the diversity of
MG-GPO is even lower than NSGA-II. It still leads to high efficiency because
the MG-GPO solutions are selected with the posterior model and are not
entirely random.
Since MG-GPO and PSO are not sensitive to noise and are more efficient than genetic algorithms, they are preferred for problems where stochastic
algorithms are required to search the global optima.
0
100
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500
evaluations
10
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0
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y = f(x)
RCDS, N=500
=0.001
=0.01
=0.1
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100
200
300
400
500
evaluations
10
-4
10
-2
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y = f(x)
RSimplex, N=500
=0.001
=0.01
=0.1
0
100
200
300
400
500
evaluations
10
-4
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y = f(x)
N-M Simplex, N=500
=0.001
=0.01
=0.1
0
50
100
150
200
250
300
evaluations
10
0
10
1
10
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y = f(x)
GP, N=300
=0.001
=0.01
=0.1
Figure 7.14 The convergence histories over 500 function evaluations of RCDS (top
left), RSimplex (top right), and N-M simplex (bottom left) and over 300 evaluations
for GP (bottom right) for the Rosenbrock-4 function with three noise levels.
level (σ = 0.001), all algorithms converge toward the Pareto front, although
the NSGA-II front is narrow and incomplete. With higher noise (σ = 0.01
or 0.1), the NSGA-II method fails to converge to the Pareto front, while the
PSO and MG-GPO algorithms converge to the same front for the medium or
high noise levels.
The Figure 7.16 left plot shows the minimum values for the two objective
functions over the course of optimization for the three algorithms for the lownoise case (σ = 0.001). The convergence speed of MG-GPO is much faster
than PSO, which is in turn much faster than NSGA-II. The cause of the slow
convergence for NSGA-II is the low diversity of its new trial solutions. The
diversity can be measured by the distribution of the crowding distance, here
defined as the distance of a solution to its nearest neighbor in the parameter
space among all previous solutions. The right plot shows the distribution of
the crowding distances for the three algorithms. Interestingly, the diversity of
MG-GPO is even lower than NSGA-II. It still leads to high efficiency because
the MG-GPO solutions are selected with the posterior model and are not
entirely random.
Since MG-GPO and PSO are not sensitive to noise and are more efficient than genetic algorithms, they are preferred for problems where stochastic
algorithms are required to search the global optima.