mathe iii lecture 8 mathe iii lecture 8. 2 constrained maximization lagrange multipliers at a...
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3 Constrained Maximization Lagrange Multipliers Intuition x y iso- f curves f(x,y) = K assume +TRANSCRIPT
Mathe IIILecture 8
2
Constrained Maximization
Lagrange Multipliers
max f(x, y) s.t. g(x, y) = c
(x, y, ) = f(x, y) - g(x, y) - c L
At a maximum point of the original problem
the derivatives of the Lagrangian vanish (w.r.t. all variables).
3
Constrained Maximization Lagrange Multipliers
Intuition max f(x, y) s.t. g(x, y) = c
x
yiso- f curves
f(x,y) = K
56
205
20assume +
4
Constrained Maximization Lagrange Multipliers
Intuition max f(x, y) s.t. g(x, y) = c
x
y f x, y = K
x yf x, y + f x, y y = 0
y = y(x)
x
y
f x, yy = -
f x, y
5
Constrained Maximization Lagrange Multipliers
Intuition max f(x, y) s.t. g(x, y) = c
x
y f x, y = K
x yf x, y + f x, y y = 0
y = y(x)
x
y
f x, yy = -
f x, y
6
Constrained Maximization Lagrange Multipliers
Intuition max f(x, y) s.t. g(x, y) = c
x
y
max f(x, y) s.t. g(x, y)= c
x yg x, y + g x, y y = 0
x x
y y
g x, y f x, y- -
g x, y f x, y
x
y
g x, yy = -
g x, y
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Constrained Maximization Lagrange Multipliers
Intuition max f(x, y) s.t. g(x, y) = c
x
y
x x
y y
g x, y f x, y- -
g x, y f x, y
y x
y x
f x, y f x, yg x, y g x, y
λ =
8
Constrained Maximization Lagrange Multipliers
Intuition max f(x, y) s.t. g(x, y) = c
y x
y x
f x, y f x, yλ =
g x, y g x, y
x xf x, y - λg x, y = 0 y yf x, y - λg x, y = 0
(x, y, ) = f(x, y) - g(x, y) - c L
g(x, y) = c
A stationary point of the Lagrangian
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Constrained MaximizationThe general case
max ........
1 1
1 2 n
m m
g (x) = cf(x , x , ..x ) s.t.
g (x) = c
j 1 m m+1 n jg x , ..x , x , ..x = c
can be explicitly expressedas functions of
1 m
m+1 n
x , ...xx , ...x
1 m+1 n m m+1 nx x , ..x , ....., x x , ..x
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Constrained Maximization The general case j 1 m m+1 n jg x , ..x , x , ..x = c
m
j jh
h=1 h s s
g gx = 0x x x
differentiating w.r.t. xs , s = m+1,…,n
j = 1, ....,ms = m + 1, ....,n
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Constrained Maximization The general casej = 1, ....,m
m
j jh
h=1 h s s
g gx = 0x x x
.....
.....
1
m m m
m1
1 s m s s
m1
1 s m s s
1 1g xx g g = 0x x x x x
.......g g gxx = 0x x x x x
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Constrained Maximization The general case
s = m + 1, ....,n
1 1
s s
m m
s s
x gx x
G ... ... 0x gx x
,
jm m j,h
h
gG G
xx.....
.....
1
m m m
m1
1 s m s s
m1
1 s m s s
1 1g xx g g = 0x x x x x
.......g g gxx = 0x x x x x
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Constrained Maximization The general case
s = m + 1, ....,n
1 1
s s-1
m m
s s
x gx x... G ...x gx x
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Constrained Maximization The general case 1 m+1 n m m+1 n m+1 nf x x , ..x , ....x x , ..x , x , ..x
maxm+1 n
m+1 nx ,..x 1 mf x , . x ,..x , ..x.
The derivatives w.r.t. xm+1,…..xn are zero at a max (min) point.
s = m + 1, ....,n
mh
h=1 h s s
xf f = 0x x x
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Constrained Maximization The general case
,...,
1
s
1 m sm
s
xx
f f f = 0x x x
xx
s = m + 1, ....,n
m
h
h=1 h s s
xf f = 0x x x
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Constrained Maximization The general case
,...,
1
s
1 m sm
s
xx
f f f = 0x x x
xx
s = m + 1, ....,n
But:
1 1
s s-1
m m
s s
x gx x... G ...x gx x
,...,
1
s-1
1 m sm
s
gx
f f fG ... = 0x x x
gx
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Constrained Maximization The general case
,...,
-1
1
s
m
s
1 m s
gx
f..f f Gx x
. = 0x
gx 1 mλ , ....λ s = m + 1, ....,n
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Constrained Maximization The general case
,...,
-1
1
s
m
s
1 m s
gx
f..f f Gx x
. = 0x
gx 1 mλ , ....λ
1
s
sm
s
1 mλ , ...
gx
f..
x
.λ . = 0x
gs = m + 1, ....,n
We need to show this for s = 1,….m
,...,
-1
1 m1 m
f fλ , ...λ Gx x
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Constrained Maximization The general case ,...,
-11 m
1 m
f fλ , ...λ Gx x
,...,
-1
1 m1 m
f fλ , ...λ G Gx x
G ,..., 1 m
f f=x x
,...,
1 m
1 m
f fλ , ...λ Gx x
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Constrained Maximization The general case ,...,
1 m1 m
f fλ , ...λ Gx x
1
s
1 ms
m
s
gx
fλ , ....λ ... = 0x
gx s = ,m + 1,1, ..,m ...,n
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Constrained Maximization The general case
, ,..., ,..., m
1 m 1 n 1 n j j 1 n jj=1
λ , ..., λ x x = f(x x ) - λ g x , ..., x - cL
1
s
1 ms
m
s
gx
f λ , ....λ ... = 0x
gx
s = ,m + 1,1, ..,m ...,n
max ........
1 1
1 2 n
m m
g (x) = cf(x , x , ..x ) s.t.
g (x) = cdefine:
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Constrained Maximization
Interpretation of the multipliers
max ........
1 1
1 2 n
m m
g (x) = cf(x , x , ..x ) s.t.
g (x) = c
Let , be the solution* * * *1 n 1 mx , ..., x λ , ..., λ
,* *i 1 m j 1 mx c , ...,c λ c , ...,c
:define * *1 nf c = f x c , ..., x c
1 mc = c , ...,c
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Constrained Maximization
Interpretation of the multipliers
* *mj
j=1i j i
f x xf c=
c x c
* *1 nf c = f x c , ..., x c
But:
* *m
h*h
h=1j j
f x g xλ
x x
* *m mh j*
hj=1 h=1i j i
g x xf c= λ
c x c
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Constrained Maximization
Interpretation of the multipliers
* *m m
h j*h
j=1 h=1i j i
g x xf c= λ
c x c
* *m mh j*
hh=1 j=1i j i
g x xf c= λ
c x c
* *m
h j
j=1 j
m*h
h 1 i=i
g x xx
f c=
c cλ
when or
* **m
h hj
j=1 j i i
g x g xx=
x c c
= 0 h i 1 h = i
25
Constrained Maximization
Interpretation of the multipliers
* *mh j
j=1 j
m*h
h 1 i=i
g x xx
f c=
c cλ
*i
i
f c= λ
c9