EAS6939 Homework #4
1. Consider the following design optimization problem:
22
12
1
11
3
32
1
Minimize
( )
4
4
Subject to
( )
0
()
0
(
1
) 0
fx
x
x
gx
x
=
+− +
=− ≤
=
−− ≤
x
x
x
x
(i)
Find the optimum point graphically
(ii)
Show that the optimum point does not satisfy KT condition. Explain
(i)
As shown in the figure, (1, 0) is the optimum point and
f
= 1 at the optimum point.
x
*
1.0
∇
g
3
(
x
*)
∇
g
2
(
x
*)
∇
f
(
x
*)
x
1
x
2
f
=1
f
=4
(ii) The Lagrangian function for the problem can be defined as
2
2
1
1
1
1
2
3
2
1
3
44(
)(
)[(
1)
]
Lx x x
x s
x
λλ
λ
=+− ++−+ + −+ +
−− +
The KT condition is
3
3
1
223
2
2
21
3
24
3
(
0
20
0
0
(1
)
0
0,
1,2,3
ii
xx
x
xs
s
si
−
=
−+
=
=
=
=
==
At
x
= (1, 0) since g
1
is inactive, and g
2
and g
3
are active, the slack variables should be
123
0,
0,
0
ss
===
The first equation in the KT condition can’t be satisfied by substituting into these values.
As is clear from the figure, the gradients of two active constraints are not independent: [0, 1]
and [0, 1]. In the mathematical term,
x
* is not a
regular point
of the feasible domain. The KT
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Optimization, X1, g2, hessian matrix

Click to edit the document details