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NMinimize
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Optimizing a Path to Avoid Obstacles
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Find a path through circular obstacles such that the distance between the start and end points is minimized.
Define the positions and radii of the obstacles and the start and end points of the path. Visualize the setup:
I
n
[
1
]
:
=
{
G
r
a
p
h
i
c
s
[
{
P
i
n
k
,
D
i
s
k
[
]
}
]
,
G
r
a
p
h
i
c
s
[
{
E
d
g
e
F
o
r
m
[
T
h
i
c
k
]
,
P
i
n
k
,
D
i
s
k
[
]
}
]
,
G
r
a
p
h
i
c
s
[
{
E
d
g
e
F
o
r
m
[
D
a
s
h
e
d
]
,
P
i
n
k
,
D
i
s
k
[
]
}
]
,
G
r
a
p
h
i
c
s
[
{
E
d
g
e
F
o
r
m
[
D
i
r
e
c
t
i
v
e
[
T
h
i
c
k
,
D
a
s
h
e
d
,
B
l
u
e
]
]
,
P
i
n
k
,
D
i
s
k
[
]
}
]
}
I
n
[
2
]
:
=
p
=
{
{
2
,
2
}
,
{
7
,
6
}
,
{
3
,
5
}
,
{
4
,
8
}
,
{
8
,
9
}
,
{
6
,
2
.
5
}
}
;
r
=
{
1
,
1
.
3
,
1
.
5
,
1
,
0
.
7
,
1
.
5
}
;
{
s
t
a
r
t
,
e
n
d
}
=
{
{
0
,
0
}
,
{
1
0
,
1
0
}
}
;
d
o
m
a
i
n
=
G
r
a
p
h
i
c
s
[
{
E
d
g
e
F
o
r
m
[
D
i
r
e
c
t
i
v
e
[
T
h
i
c
k
,
B
l
u
e
]
]
,
L
i
g
h
t
e
r
[
B
l
u
e
,
0
.
8
]
,
,
M
a
p
T
h
r
e
a
d
[
D
i
s
k
[
#
1
,
#
2
]
&
,
{
p
,
r
}
]
,
{
P
o
i
n
t
S
i
z
e
[
0
.
0
2
]
,
P
o
i
n
t
[
{
s
t
a
r
t
,
e
n
d
}
]
}
}
,
F
r
a
m
e
T
r
u
e
]
O
u
t
[
2
]
=
The path is discretized into
n
points with distance of
l
e
n
/
n
between points, where
l
e
n
is the trajectory length being minimized:
I
n
[
3
]
:
=
n
=
7
5
;
d
i
s
t
a
n
c
e
C
o
n
s
t
r
a
i
n
t
s
=
T
a
b
l
e
[
I
n
a
c
t
i
v
e
[
N
o
r
m
]
[
x
[
i
]
-
x
[
i
-
1
]
]
≤
l
e
n
/
n
,
{
i
,
n
}
]
;
The points cannot be inside the circular objects:
I
n
[
4
]
:
=
o
b
j
e
c
t
C
o
n
s
t
r
a
i
n
t
s
=
T
a
b
l
e
[
N
o
r
m
[
-
p
〚
j
〛
+
x
[
i
]
]
≥
r
〚
j
〛
,
{
i
,
1
,
n
}
,
{
j
,
1
,
6
}
]
;
The start and end points are known:
I
n
[
5
]
:
=
p
o
s
i
t
i
o
n
C
o
n
s
t
r
a
i
n
t
s
=
{
x
[
0
]
s
t
a
r
t
,
x
[
n
]
e
n
d
}
;
Collect the variables:
I
n
[
6
]
:
=
v
a
r
s
=
A
p
p
e
n
d
[
T
a
b
l
e
[
x
[
i
]
∈
V
e
c
t
o
r
s
[
2
,
R
e
a
l
s
]
,
{
i
,
0
,
n
}
]
,
l
e
n
∈
R
e
a
l
s
]
;
Minimize the length
l
e
n
subject to the constraints:
I
n
[
7
]
:
=
S
h
o
r
t
[
r
e
s
=
N
M
i
n
i
m
i
z
e
[
{
l
e
n
,
d
i
s
t
a
n
c
e
C
o
n
s
t
r
a
i
n
t
s
,
o
b
j
e
c
t
C
o
n
s
t
r
a
i
n
t
s
,
p
o
s
i
t
i
o
n
C
o
n
s
t
r
a
i
n
t
s
}
,
v
a
r
s
,
M
e
t
h
o
d
"
D
i
f
f
e
r
e
n
c
e
O
f
C
o
n
v
e
x
"
]
,
2
]
O
u
t
[
7
]
/
/
S
h
o
r
t
=
{
1
4
.
5
8
4
2
,
{
x
[
0
]
{
4
.
0
6
3
5
9
×
-
9
1
0
,
4
.
0
3
1
9
9
×
-
9
1
0
}
,
x
[
1
]
{
0
.
1
7
7
1
4
9
,
0
.
0
8
0
1
9
8
2
}
,
x
[
2
]
{
0
.
3
5
4
2
9
7
,
0
.
1
6
0
3
9
6
}
,
x
[
3
]
{
0
.
5
3
1
4
4
6
,
0
.
2
4
0
5
9
5
}
,
x
[
4
]
{
0
.
7
0
8
5
9
4
,
0
.
3
2
0
7
9
3
}
,
6
8
,
x
[
7
3
]
{
9
.
7
2
4
0
6
,
9
.
7
2
5
9
4
}
,
x
[
7
4
]
{
9
.
8
6
2
0
3
,
9
.
8
6
2
9
7
}
,
x
[
7
5
]
{
1
0
.
,
1
0
.
}
,
l
e
n
1
4
.
5
8
4
2
}
}
Visualize the result:
I
n
[
8
]
:
=
p
t
s
=
v
a
r
s
〚
1
;
;
-
2
,
1
〛
/
.
r
e
s
〚
2
〛
;
S
h
o
w
[
d
o
m
a
i
n
,
G
r
a
p
h
i
c
s
[
{
R
e
d
,
P
o
i
n
t
[
p
t
s
]
}
]
]
O
u
t
[
8
]
=
See Also
AntColonyOptimization
MaximizeOverPermutations
MinSumPermutation
Related Symbols
NMinimize
NMaximize
Minimize
Maximize
KnapsackSolve
FindMinimum
FindMaximum
FindFit
ConvexOptimization
Publisher Information
Contributed by:
Paco Jain (Wolfram Research)
