Lagrange Multipliers
Sometimes we need to optimize functions subject to some constraint.
To solve such constrained optimization problems we use the
method of Lagrange Multipliers
To optimize
subject to
:
)
,
(
y
x
f
0
)
,
(
=
y
x
g
1)
Write the new function
=+
)
,
,
(
λ
y
x
F
)
,
(
y
x
g
,
(
y
x
f
)
2) Find
F
,
, and
and set them all equal to zero.
Solve this system to find the critical points.
x
y
F
F
To solve this system it is often helpful to solve
and
for
.
Set those two expressions for
equal to each other
and use the result with
to find x and y.
0
=
x
F
0
=
y
F
0
=
F
3)
The solution to the original problem (if it exists) will occur at one of these critical points
Note:
This method only finds the critical points, it does not tell whether the function is maximized, minimized or neither at the critical
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 Fall '05
 PamelaSatterfield
 Optimization, Mathematical optimization, lagrange multipliers, Constraint satisfaction

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