tributed uniformly over [ − 4,4]
n
, n = 2, 5, and 10 which is symmetric about the optimum solu-
tion.
Fig. 8. The PI for the GMO, CMO, and MMO across the settings of the step size, s, for the 10-dimensional sphere
function F
1
. For all three operators, the PI values decrease with increasing sigma. Since the quadratic function is continu-
ous and unimodal, the PI value attains a peak value of 0.5 as s tends to 0, which is the PI value on an inclined plane. The PI
curve for the CMO drops the fastest and is followed by those for the MMO and GMO. Paralleling the estimates for EI Fig.
7, the GMO offers the greatest PI for any fixed value of s, followed by MMO and CMO, respectively. For s 0 the
PI 0.5, and as s becomes large the PI tends to zero. Fig. 6. The EI for the GMO, CMO, and MMO across the
settings of the step size, s, for the 2-dimensional sphere func- tion F
1
. The CMO EI curve peaks first, followed by those for the GMO and MMO. The maximum EI occurs at s = 0.47,
0.49, and 0.21 for the GMO, MMO, and CMO, respectively. The corresponding peak EI values were 0.20, 0.19, and 0.20
for the GMO, MMO, and CMO, respectively. The GMO curve has the largest bandwidth, followed by those for the
MMO and CMO.
4. Results
Figs. 6 and 7 show the mean EI as a function of s for the GMO, CMO, and MMO on the 2- and
10-dimensional sphere F
1
. Certain similar pat- terns are evidenced immediately. First, in both
cases, the GMO offers a greater peak EI than does the MMO or CMO. In addition, the peak EI
occurs for GMO, MMO, and CMO with decreas- ing values of s. That is, larger s values are re-
quired to generate the peak EI for GMO than are required for either MMO or CMO. As expected,
for s 0 the EI 0, and as s becomes large the EI turns negative i.e. the typical step size of the
variation operator is larger than twice the distance to the optimum and the resulting offspring are
worse than their parent. Fig. 8 shows the PI for each operator on the 10-dimensional F
1
. Parallel- ing the estimates for EI, the GMO offers the
greatest PI for any fixed value of s, followed by MMO and CMO, respectively. For s 0 the PI
0.5, and as s becomes large the PI tends to zero.
Of particular interest is the corresponding rela- tionship between PI and EI for each operator on
Fig. 7. The EI for the GMO, CMO, and MMO across the settings of the step size, s, for the 10-dimensional sphere
function. As in the 2-dimensional case Fig. 6, the CMO EI curve peaks first, followed by those for the MMO and GMO.
The maximum EI occurs at s = 1.01, 0.69, and 0.27 for the GMO, MMO, and CMO, respectively. The corresponding
peak EI values were 9.43, 8.01, and 7.79 for the GMO, MMO, and CMO, respectively. The GMO curve has the largest
bandwidth BW, followed by those for the MMO and CMO.
Fig. 9. The relationship between the PI and EI for the GMO, CMO, and MMO on the 10-dimensional sphere function F
1
. For very small PI there is a corresponding negative EI regard-
less of the variation operator, but as PI increases there is a wide range of values that correspond to essentially similar
values of EI.
there is a wide range of values that correspond to essentially similar values of EI. Table 1 shows the
PI associated with the peak EI for each operator for the 10-dimensional F
1
and other functions. As indicated, regardless of the variation operator
the best choice for PI is considerably less than 0.2 cf. the 15 rule.
One method for assessing the robustness of a particular variation operator concerns the range
of values for s that will yield reasonable values of EI. For the case of unimodal fitness distribution
curves, define the bandwidth to be the range of values s such that
EIs \ 0.5 EI 21
where EIs is the expected improvement for a scaling value s and EI is the peak EI. Table 2
indicates that the bandwidth for the GMO was almost twice as wide as for the CMO. In this
sense, the GMO is less sensitive to particular values of s than the CMO.
In a similar manner, Figs. 10 and 11 offer the results for the 10-dimensional Ackley and Rast-
the 10-dimensional F
1
see Fig. 9. For very small PI there is a corresponding negative EI regardless
of the variation operator, but as PI increases
Table 1 The scale factor s, expected improvement EI, and probability of improvement PI values at the peaks of the EI curves as a
function of the step size s for the test function F
1
–F
4
EI peak values PI value when EI peaks
s at EI peak s
p
EI peak value EI
p
Dim = 10 GMO
CMO GMO
CMO GMO
MMO MMO
CMO MMO
F
1
1.01 0.15
Sphere 0.10
0.69 0.15
7.79 8.01
9.43 0.27
1.01 F
2
Ackley 0.14
0.13 0.15
1.01 1.22
0.25 0.53
0.99 203.94
180.35 175.48
0.13 Rastrigin
0.12 F
4
0.13 1.05
0.63 0.33
2.88 Step
2.51 F
4
2.47 0.09
0.08 0.09
0 99
0.55 0.27
Table 2 The left and right EI bandwidths for the test functions F
1
–F
4 a
Overall EI bandwidth Left half EI bandwidth
EI peak values Right half EI bandwidth
Dim = 10 GMO
MMO CMO
GMO MMO
CMO GMO
MMO CMO
F
1
0.74 0.66
0.26 0.98
0.92 0.68
Sphere 1.72
1.58 0.94
F
2
0.82 0.50
0.24 0.92
Ackley 0.98
0.64 1.74
1.48 0.88
Rastrigin F
4
0.86 0.60
0.32 1.00
1.04 0.68
1.86 1.64
1.00 1.74
0.88 0.62
1.48 Step
F
4
0.80 0.52
0.26 0.94
0.96
a
The bandwidths defined in the text or GMO are almost twice as large as those for the MMO.
rigin functions. Despite the presence of multiple local minima in these functions, the EI and PI
curves appear essentially similar to those obtained for F
1
. GMO offers a greater peak EI and has a larger bandwidth than MMO or CMO. Interest-
ingly, for the step function F
4
, Fig. 12 shows that EI is not a function of PI i.e. it is one-to-many.
This illustrates that assessing the appropriate value of PI alone may not be sufficient to maxi-
mize EI. Note also that the maximum PI for this case never exceeds 0.2, thus it would be impossi-
ble to apply the 15 rule to this problem.
5. Discussion
