Online optimization algorithms 201
7.3.4 Performance comparison for single-objective algorithms
The tests with function f 1 (x) in Eq. (7.31) demonstrate the impact of noise on
the performances of a few selected single-objective optimization algorithms.
Figure 7.12 summarizes the test results. It shows the comparison of the sorted
minimum function values found in 300 evaluations for 100 runs by the algorithms under three noise levels. Under low noise (σ = 0.001), almost all algorithms can locate the minimum (with final minimum f min < 0.1), except the
N-M simplex method occasionally fails. Under medium noise (σ = 0.01), the
gradient descent method and the N-M simplex often fail to achieve f min < 0.1,
while the others usually can. When the noise level is high (σ = 0.1), all other
algorithms fail to reach f min < 0.1, except for the RCDS method. The RCDS
is the least sensitive to noise among all tested algorithms. The RSimplex
method outperforms the original simplex in all three noise levels, although it
falls behind the RCDS method.
To further characterize the performances of the algorithms, additional tests
are done with the Rosenbrock function [100]. The Rosenbrock function is a
non-convex function, given in the form
f 2 (x) =
N −1
i=1
100(x i+1 − x
2
i )
2 + (1 − x i )
2
,
(7.34)
where N is the number of dimensions in x. In the test we choose N = 4,
in which case the Rosenbrock function has one global minimum located at
x m = (1, 1, 1, 1)
T , with f 2 (x m ) = 0, and a local minimum at (−1, 1, 1, 1)
T
with the function value of 4. The parameter ranges are chosen to be the same
as the tests for the function in Eq. (7.31), with x i ∈ [−2, 2] for i = 1-4. The
initial solution is also chosen to be the same, with x 0 = (−0.5, 0, −0.5, 0)
T ,
where the function value is f (x 0 ) = 43.
The tests are run with three noise levels, with σ = 0.001, 0.01, and 0.1.
The number of evaluations is limited to 500 for all algorithms except for the
GP optimizer. For the GP the number of evaluations is limited to 300 as it
becomes too time consuming and there is no clear gain with more evaluations
(see Figure 7.14). The final minimum function values achieved in 100 runs are
sorted and plotted in Figure 7.13 for all algorithms and the three noise levels.
With low noise (σ = 0.001), the RSimplex and N-M simplex methods have
similar performance, and both of them outperform the other methods. With
medium noise (σ = 0.01), the RSimplex shows better resistance to noise than
the N-M simplex method. The RCDS method also performs better than the
original simplex method for most of the cases. At a high noise level (σ = 0.1),
while RSimplex is still better than the original simplex method, it is not as
good as the RCDS method. The performance of the RCDS method for the
three noise levels was similar, but its efficiency was not high. This is due to the
nature of the Rosenbrock function, for which the valley in the parameter space
leading to the minimum is banana-shaped, such that the direction toward the
minimum is constantly changing. Figure 7.14 shows the convergence histories
7.3.4 Performance comparison for single-objective algorithms
The tests with function f 1 (x) in Eq. (7.31) demonstrate the impact of noise on
the performances of a few selected single-objective optimization algorithms.
Figure 7.12 summarizes the test results. It shows the comparison of the sorted
minimum function values found in 300 evaluations for 100 runs by the algorithms under three noise levels. Under low noise (σ = 0.001), almost all algorithms can locate the minimum (with final minimum f min < 0.1), except the
N-M simplex method occasionally fails. Under medium noise (σ = 0.01), the
gradient descent method and the N-M simplex often fail to achieve f min < 0.1,
while the others usually can. When the noise level is high (σ = 0.1), all other
algorithms fail to reach f min < 0.1, except for the RCDS method. The RCDS
is the least sensitive to noise among all tested algorithms. The RSimplex
method outperforms the original simplex in all three noise levels, although it
falls behind the RCDS method.
To further characterize the performances of the algorithms, additional tests
are done with the Rosenbrock function [100]. The Rosenbrock function is a
non-convex function, given in the form
f 2 (x) =
N −1
i=1
100(x i+1 − x
2
i )
2 + (1 − x i )
2
,
(7.34)
where N is the number of dimensions in x. In the test we choose N = 4,
in which case the Rosenbrock function has one global minimum located at
x m = (1, 1, 1, 1)
T , with f 2 (x m ) = 0, and a local minimum at (−1, 1, 1, 1)
T
with the function value of 4. The parameter ranges are chosen to be the same
as the tests for the function in Eq. (7.31), with x i ∈ [−2, 2] for i = 1-4. The
initial solution is also chosen to be the same, with x 0 = (−0.5, 0, −0.5, 0)
T ,
where the function value is f (x 0 ) = 43.
The tests are run with three noise levels, with σ = 0.001, 0.01, and 0.1.
The number of evaluations is limited to 500 for all algorithms except for the
GP optimizer. For the GP the number of evaluations is limited to 300 as it
becomes too time consuming and there is no clear gain with more evaluations
(see Figure 7.14). The final minimum function values achieved in 100 runs are
sorted and plotted in Figure 7.13 for all algorithms and the three noise levels.
With low noise (σ = 0.001), the RSimplex and N-M simplex methods have
similar performance, and both of them outperform the other methods. With
medium noise (σ = 0.01), the RSimplex shows better resistance to noise than
the N-M simplex method. The RCDS method also performs better than the
original simplex method for most of the cases. At a high noise level (σ = 0.1),
while RSimplex is still better than the original simplex method, it is not as
good as the RCDS method. The performance of the RCDS method for the
three noise levels was similar, but its efficiency was not high. This is due to the
nature of the Rosenbrock function, for which the valley in the parameter space
leading to the minimum is banana-shaped, such that the direction toward the
minimum is constantly changing. Figure 7.14 shows the convergence histories