App endix
The app endix provides the description of 11 test cases,
G
1{
G
11 for constrained parameter opti-
mization problems.
Minimize
G
1(
~x
)=5
x
1
+5
x
2
+5
x
3
+5
x
4
5
P
4
i
=1
x
2
i
P
13
i
=5
x
i
,
sub ject to the following constraints:
2
x
1
+2
x
2
+
x
10
+
x
11
10, 2
x
1
+2
x
3
+
x
10
+
x
12
10, 2
x
2
+2
x
3
+
x
11
+
x
12
10,
8
x
1
+
x
10
0,
8
x
2
+
x
11
0,
8
x
3
+
x
12
0,
2
x
4
x
5
+
x
10
0,
2
x
6
x
7
+
x
11
0,
2
x
8
x
9
+
x
12
0,
and bounds 0
x
i
1,
i
=1
;:::;
9, 0
x
i
100,
i
=10
;
11
;
12, 0
x
13
1.
The problem has 13 variables and 9 linear constraints; the function
G
1 is quadratic with its global
minimum at
~x
=(1
;
1
;
1
;
1
;
1
;
1
;
1
;
1
;
1
;
3
;
3
;
3
;
1),
where
G
1(
~x
)=
15. Six (out of nine) constraints are active at the global optimum (all except the
following three:
8
x
1
+
x
10
0,
8
x
2
+
x
11
0,
8
x
3
+
x
12
0).
Maximize
G
2(
~x
)=
j
P
n
i
=1
cos
4
(
x
i
)
2
Q
n
i
=1
cos
2
(
x
i
)
p
P
n
i
=1
ix
2
i
j
;
sub ject to
Q
n
i
=1
x
i
0
:
75 ,
P
n
i
=1
x
i
7
:
5
n
, and bounds 0
x
i
10 for 1
i
n
.
Function
G
2 is nonlinear and its global maximum is unknown. For
n
= 20, solutions
~x
for which
G
2(
~x
)=0
:
8036 were reported.
Maximize
G
3(
~x
)=(
p
n
)
n
Q
n
i
=1
x
i
,
where
P
n
i
=1
x
2
i
=1 and 0
x
i
1 for 1
i
n
.
The function
G
3 has a global solution at (
x
1
;:::;x
n
)=(
1
p
n
;:::;
1
p
n
) and the value of the function
in this point is 1.
Minimize
G
4(
~x
)=5
:
3578547
x
2
3
+0
:
8356891
x
1
x
5
+37
:
293239
x
1
40792
:
141,
sub ject to three double inequalities:
0
85
:
334407 + 0
:
0056858
x
2
x
5
+0
:
0006262
x
1
x
4
0
:
0022053
x
3
x
5
92
90
80
:
51249 + 0
:
0071317
x
2
x
5
+0
:
0029955
x
1
x
2
+0
:
0021813
x
2
3
110
20
9
:
300961 + 0
:
0047026
x
3
x
5
+0
:
0012547
x
1
x
3
+0
:
0019085
x
3
x
4
25,
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