Solutions of Homework #5, 1998 Fall
MEAM 501
Analytical Methods in Mechanics and Mechanical Engineering
1.
Consider a constrained minimization problem
( 29 ( 29
n
n
n
n
T
T
K
F
F
R
b
R
A
R
v
b
v
Av
v
v
v
v
∈
∈
∈

=
×
∈
,
,
,
2
1
,
min
{ }
m
n
m
n
K
R
g
R
B
0
g
Bv
R
v
∈
∈
≤

∈
=
×
,
,
:
where a matrix
A
is symmetric, that is,
A
A
=
T
, and the constrained set
K
is non
empty.
(1)
Find the necessary condition that an element
K
∈
u
is a minimizer of the
functional F on the constrained set
K
.
( 29( 29
( 29
( 29
( 29 ( 29
K
F
F
T
T
∈
2200
≥



=

+
=

→
v
b
u
v
Av
u
v
u
v
u
u
v
u
,
0
lim
0
a
a
d
a
(2)
Obtain the Lagrangian
L
to this constrained minimization problem, and set up the
"equivalent" unconstrained problem on the primal variable
v
by considering the
necessary condition of the problem obtained by the Lagrange multiplier method.
Lagrangian L is defined by
( 29 ( 29 ( 29
( 29
0
R
g
Bv
b
v
Av
v
g
Bv
v
v
≤
∈



=


=
m
T
T
T
T
F
L
,
2
1
,
and we shall consider the problem
( 29
v
0
v
,
max
min
L
≤
Suppose that
( 29
0
R
R
u
≤
×
∈
,
,
m
n
is a solution of the minmax problem. Then we
have
(
1
)
( 29 ( 29
n
L
L
R
v
v
u
∈
2200
≤
,
,
,
(
2
)
( 29 ( 29
0
R
u
u
≤
∈
2200
≤
m
L
L
,
,
,
From (1), we have the following necessary condition
( 29( 29
( 29
( 29 ( 29 ( 29
n
T
T
L
L
R
v
b
B
Au
u
v
u
v
u
u
v
u
∈
2200
≥



=

+
∂
∂
=

→
,
0
,
lim
,
0
a
a
d
a
Since
v
is arbitrary, and since we can take
u
u
v
d
±
=
, we have
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
n
T
T
R
u
b
B
Au
u
∈
2200
≥


±
d
d
,
0
that is
n
T
in
R
b
B
Au
=

Similarly, from the second inequality relation (2), we have
( 29( 29
( 29
( 29 ( 29 ( 29
0
R
g
Bu
u
u
≤
∈
2200
≤



=
+
∂
∂
=

→
m
T
L
L
,
0

,
lim
,
0
a
a
d
a
that is
( 29 ( 29
0
R
g
Bu
≤
∈
2200
≥


m
T
,
0
Combining these, the necessary condition of the Lagrange multiplier formulation
becomes
(
a
)
n
T
in
R
b
B
Au
=

(
b
)
( 29 ( 29
0
R
g
Bu
≤
∈
2200
≥


m
T
,
0
The second inequality yields also the KKT condition
( 29
0
,
,
=

≤
≤

g
Bu
0
0
g
Bu
T
(3) Solve the problem for
A
,
B
, and
g
obtained by the following MATLAB program.
=


















=
1
1
1
1
1
1
1
1
1
1
,
2
1
0
0
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
0
0
1
2
b
A
=
=
8
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '09

Click to edit the document details