Geoscience Reference
In-Depth Information
point in the simplex ( X b ), with the aim of directing the simplex toward the
optimum using a lesser number of function evaluations. Thus, not only the
worst point, but also the best point in a simplex is used, making better use
of the already available information.
The reflected and contracted points are shifted toward the best point
using a parameter theta, θ , which is defined as below for reflection and
contraction, respectively.
X new = ((1 . 0
θ )
X ref )+( θ
X b ) ,
(1)
X new = ((1 . 0
θ )
X con )+( θ
X b ) ,
(2)
where X ref is the reflected point, X con the contracted point and X b is the
best point in the simplex. The parameter θ can take values between 0.0
and 1.0. The higher its value, the more is the exploitation pressure, since
the new point ( X new ) moves closer to the best point ( X b ). For θ =0 . 5, the
new point, X new is in the middle of the reflected (or contracted) point and
the best point ( X b ), which is shown in Figs. 2(a) and (b), for reflection and
contraction steps, respectively.
4. Experiment on Test Functions
In this section, a performance comparison between the original SCE-UA
and the enhanced SCE-UA on a series of test functions is presented.
For the two types of SCE-UA algorithms, performance criteria used are:
(i) the number of failures (NF) out of 100 trials; and (ii) the average number
of function evaluations (AFE) resulting from successful trials. NF measures
robustness while AFE describes the eciency of the algorithm.
The stopping criteria used are as follows. A trial is deemed a success as
soon as the best function value in the sample became less than 10 3 .How-
ever, if the trial reached 25,000 function evaluations without reducing the
best function value below 10 3 , the trial was deemed a failure. Exception
to the stopping criterion of 25,000 function evaluations are the Neumaier
No. 3 function and the Griewank 10D function, which being 10-dimensional
functions, are expected to require higher number of function evaluations.
As such, the maximum function evaluation for these two test functions is
set to 50,000.
The results of the comparison are presented in Table 1. For the enhanced
version of the algorithm, the value of θ that gave the best results is also
presented in Table 1. It is seen that best results are obtained when θ is
Search WWH ::




Custom Search