May 2015
Intermediate to advanced
466 pages
10h 33m
English
Content preview from Optimization
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194 Optimization: Algorithms and Applications
subject to
g
1
(x) = 2x
1
− x
2
≤ 0
g
2
(x) = x
1
≤ 5
At x = (2, 1)
f(x) = 2
∇ =
−
−
=
−
f
x
x
( )
( )
( )
x
2 1
2 2
2
2
1
2
The constraint g
1
(x) is also violated. Therefore,
N =
−
2
1
The projection matrix is given by
P I N N N N= − =
−
( )
T T1
1
5
1 2
2 4
Therefore, the search direction is given by
S P x= − ∇ =
f( )
1
2
Taking γ = 0.1,
α
γ
= −
∇
=
f
f
T
( )
( )
.
x
S x
0 1
The value of x can now be updated as
x x S N N N x= + − =
+
−
−
−
α ( ) .
.
( )
T
i
g
1
2
1
0 1
1
2
1 2
0..
.
.
6
0 9
1 8
=
195Constrained Op ...
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