1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday...
-
Upload
erin-whitler -
Category
Documents
-
view
218 -
download
3
Transcript of 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday...
![Page 1: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/1.jpg)
1
![Page 2: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/2.jpg)
2
No lecture on Wed February 8th
Thursday 9th FebFriday 27th JanFriday 10th Feb
Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N
![Page 3: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/3.jpg)
3
Non Linear Programming
if is a solution of the problem*x
j j = 1,3 .λ 0,. ..,m
is a stationary point (w.r.t. ) of the Lagrangian :
-
*i
m
i i ii=1
2 x. x
x = f x λ g x -c L
*j j1. j = 1, .g x mc , ..,
if then *j j jg x < c λ = j = 1, .4. 0, ..,m
max s.t.
1 1
m m
x
g x c
g x c
f x ......
Kuhn – Tucker conditions
![Page 4: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/4.jpg)
4
Non Linear Programming
if is a solution of the problem*x
j j jλ g x - c = 0
if then *j j jg x < c λ = j = 1, .4. 0, ..,m
max s.t.
1 1
m m
x
g x c
g x c
f x ......
Kuhn – Tucker conditions
j j jλ4. j =g x - c = 1, .0, ..,m
if then *j j jg x < c λ = j = 1, .4. 0, ..,m
![Page 5: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/5.jpg)
5
Nonnegative variables
max . .s t
x
g x, y c
f x, y x 0
y 0
- - 1 2x, y = f x, y λ g x, y - c -x -yL
max . .s t
x
g x, y
-x 0
-y 0
c
f x, y
1
2
x x
y y
f (x, y) - λg (x, y)+ μ = 0x
f (x, y) - λg (x, y)+ μ = 0y
L
L1 2λ, μ , μ 0
1 2λ g x, y - c x = y = 0
![Page 6: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/6.jpg)
6
Nonnegative variables
g x, y c
x 0
y 0
1
2
x x
y y
f (x, y) - λg (x, y)+ μ = 0x
f (x, y) - λg (x, y)+ μ = 0y
L
L
1 2λ, μ , μ 0 1 2λ g x, y - c x = y = 0
1
2
x x
y y
f (x, y) - λg (x, y) = -μ 0
f (x, y) - λg (x, y) = -μ 0
1 x xx > 0 μ = 0 f (x, y) - λg (x, y) = 0
![Page 7: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/7.jpg)
7
Nonnegative variables
g x, y c
x 0
y 0
λ 0 λ g x, y - c 0
x x x x
y y y y
f (x, y) - λg (x, y) x = 0, f (x, y) - λg (x, y) 0
f (x, y) - λg (x, y) y = 0 f (x, y) - λg (x, y) 0
For a general problem:
max s.t.
1 1
1 n
m m
x
g x c
x 0 x 0
g x c
f x ...... ......
![Page 8: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/8.jpg)
8
Nonnegative variables
j j j jλ g x - c 0, λ 0, j = 1, ...m
m
j hj=1 h
jh
g (x)f (x) - λ x = 0, h = 1, ....,n
x
max s.t.
1 1
1 n
m m
x
g x c
x 0 x 0
g x c
f x ...... ......
m
jj=1 h
jh
g (x)f (x) - λ 0, h = 1, ....,n
x
![Page 9: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/9.jpg)
9
Nonnegative variables
Example: Peak Load Pricing
The price of a good (electricity) for time period i is given as pi
The producer chooses how much to produce in each period (xi), and the maximal capacity of his
plant (k).
The total cost of producing(x1,…,xn) is C(x1,…,xn).The cost of capacity k is D(k).
![Page 10: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/10.jpg)
10
Nonnegative variables
Example: Peak Load Pricing
The producer maximizes:
n
1 n i i 1 ni=1
π x , ..., x ,k = p x - C x , ..., x - D k
s.t. i 0 x k i = 1,...,n
n n
1 n i i 1 n i ii=1 i=1
x , ..., x ,k = p x - C x , ..., x - D k λ x - kL
i 1 n ii
ip - C x , ..., x - λ 0x
L
![Page 11: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/11.jpg)
11
Nonnegative variables Example: Peak Load Pricing
n n
1 n i i 1 n i ii=1 i=1
x , ..., x ,k = p x - C x , ..., x - D k λ x - kL
i 1 n ii
ip - C x , ..., x - λ 0x
L
i 1 n i ιip - C x , ..., x - λ x = 0
n
ii=1
-D k + λ 0k
L
n
ii=1
-D k + λ k 0
i i iλ 0, λ x - k 0, i = 1, ...,n
i i i 1 n ιif x 0, p C x , ...x
i i
i i 1 n
if k > x 0, λ = 0
p C x , ...x
price marginal costi.e. equals in off - peak hours.
. ipeak
for peak hours : D k = λ
![Page 12: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/12.jpg)
12
The Maximum Principle
Optimization over time
Stock – state variablesFlow – control variables
A.K. Dixit: Optimization in Economic Theory, Oxford University Press, 1989. Chapter 10
*
*
t = 1,2,3, .....,T
t+1 t t ty - y = Q y ,z ,t
stocks of capital goods
consumption, labor supplyflow variable
production function
![Page 13: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/13.jpg)
13
The Maximum Principle
Optimization over time
Stock – state variablesFlow – control variables
t = 1,2,3, .....,T
t+1 t t ty - y = Q y ,z ,t
t+1 t t ty y + Q y ,z ,t t tG y ,z ,t 0
![Page 14: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/14.jpg)
14
The Maximum Principle
Optimization over time
t+1 t t ty - y = Q y ,z ,t
t tG y ,z ,t 0 t = 0,1,2, .....,T
T
t tt=0
F y ,z ,t
additively separable utility function
F a,0 + F b,1 + F c,2
The marginal rate of substitution between periods 1,2
F c,2-
F b,1is independent of the quantitiy consumed in period 0
![Page 15: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/15.jpg)
15
The Maximum Principle
. .s t
t+1 t t t
t t
t = 0,1, .....,Ty - y = Q y ,z ,t
G y ,z ,t 0
maxT
t tt =0t t
y ,zF y ,z ,t
{
}
T
t tt=0
t+1 t t t t+1 t t t
F y ,z ,t
+ π y + Q y ,z ,t - y - λ G y ,z ,t
L
![Page 16: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/16.jpg)
16
The Maximum Principle
. .s t
t+1 t t t
t t
t = 0,1, .....,Ty - y = Q y ,z ,t
G y ,z ,t 0
maxT
t tt =0t t
y ,zF y ,z ,t
{
}
T
t tt=0
t+1 t t t t+1 t t t
F y ,z ,t
+ π y + Q y ,z ,t - y - λ G y ,z ,t
L
![Page 17: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/17.jpg)
17
.......
1 0 0 0 1 t t -1 t -1 t -1 t
t+1 t t t t+1
π y + Q y ,z ,0 - y π y + Q y ,z ,t - 1 - y
+ π y + Q y ,z ,t - y + .........
{
}
T
t tt=0
t+1 t t t t+1 t t t
F y ,z ,t
+ π y + Q y ,z ,t - y - λ G y ,z ,t
L
z t t t+1 z t t t z t tt
F y ,z ,t π Q y ,z ,t - λ G y ,z ,t = 0z
Lderivative w.r.t. zt:
derivative w.r.t. yt:
.......
1 0 0 0 1 t t -1 t -1 t -1
t+1 t tt t +1
tπ y + Q y ,z ,0 - y π y + Q y ,z ,t - 1 -
+
y
π + Q ,z ,t - y + .....y y ....
T
t+1 t t t t+1t=0
π y + Q y ,z ,t - y
y t t t+1 y t t t y t tt
t+1 t
F y ,z ,t π Q y ,z ,t - λ G y ,z ,ty
+ π - π = 0
L
![Page 18: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/18.jpg)
18
z t t t+1 z t t t z t tt
F y ,z ,t π Q y ,z ,t - λ G y ,z ,t = 0z
L
y t t t+1 y t t t y t tt
t+1 t
F y ,z ,t π Q y ,z ,t - λ G y ,z ,ty
+ π - π = 0
L
t+1 t y t t t+1 y t t t y t tπ - π = - F y ,z ,t π Q y ,z ,t - λ G y ,z ,t
Define the Hamiltonian:
H y,z,π,t = F y,z,t πQ y,z,t
maximizes s.t. t t t t+1 t tz H y ,z ,π ,t G y ,z ,t 0
Let be the value at maximum*t t+1H y ,π ,t
![Page 19: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/19.jpg)
19
z t t t+1 z t t t z t tF y ,z ,t π Q y ,z ,t - λ G y ,z ,t = 0
t+1 t y t t t+1 y t t t y t tπ - π = - F y ,z ,t π Q y ,z ,t - λ G y ,z ,t
H y,z,π,t = F y,z,t πQ y,z,t
The Hamiltonian is maximized at s.t. t t t z G y ,z ,t 0
is the value at maximum.*t t+1H y ,π ,t
The two Lagrange conditions:
The Hamiltonian:
![Page 20: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/20.jpg)
20
z t t t+1 z t t t z t tF y ,z ,t π Q y ,z ,t - λ G y ,z ,t = 0
t+1 t y t t t+1 y t t t y t tπ - π = - F y ,z ,t π Q y ,z ,t - λ G y ,z ,t
H y,z,π,t = F y,z,t πQ y,z,t
Define :
(the Lagrangian at )t t t+1 t t tL = H(y ,z ,π ,t) - λ G y ,z ,t
t
The two Lagrange conditions:
The Hamiltonian:
The Hamiltonian is maximized at s.t. t t t z G y ,z ,t 0
is the value at maximum.*t t+1H y ,π ,t
t+1 t y t t t+1π - π = -L y ,z ,π ,t
From the envelope theorem:
*t+1 t y t t+1π - π = -H y ,π ,t
![Page 21: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/21.jpg)
21
max
s.t. j
x
g x,r = 0, j = 1, ...,m
f x,r
the solution is
and the maximum is
,
*
* *
x r
f r = f x r ,r
Envelope Theorem
* **
j j
x , λ ,rf r=
r r
L
![Page 22: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/22.jpg)
22
z t t t+1 z t t t z t tF y ,z ,t π Q y ,z ,t - λ G y ,z ,t = 0
t+1 t y t t t+1 y t t t y t tπ - π = - F y ,z ,t π Q y ,z ,t - λ G y ,z ,t
H y,z,π,t = F y,z,t πQ y,z,t
Define :
(the Lagrangian at )t t t+1 t t tL = H(y ,z ,π ,t) - λ G y ,z ,t
t
The two Lagrange conditions:
The Hamiltonian:
t+1 t y t t t+1π - π = -L y ,z ,π ,t
From the envelope theorem:
*t+1 t y t t+1π - π = -H y ,π ,t
![Page 23: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/23.jpg)
23
z t t t+1 z t t t z t tF y ,z ,t π Q y ,z ,t - λ G y ,z ,t = 0
t+1 t y t t t+1 y t t t y t tπ - π = - F y ,z ,t π Q y ,z ,t - λ G y ,z ,t
H y,z,π,t = F y,z,t πQ y,z,t
Define :
(the Lagrangian at )t t t+1 t t tL = H(y ,z ,π ,t) - λ G y ,z ,t
t
The two Lagrange conditions:
The Hamiltonian:
t+1 t y t t t+1π - π = -L y ,z ,π ,t
From the envelope theorem:
*t+1 t y t t+1π - π = -H y ,π ,t
Similarly from the envelope theorem:*π πH = L = Q
![Page 24: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/24.jpg)
24
z t t t+1 z t t t z t tF y ,z ,t π Q y ,z ,t - λ G y ,z ,t = 0
H y,z,π,t = F y,z,t πQ y,z,t
Define :
(the Lagrangian at )t t t+1 t t tL = H(y ,z ,π ,t) - λ G y ,z ,t
t
The two Lagrange conditions:
The Hamiltonian:
*t+1 t y t t+1π - π = -H y ,π ,t
Similarly from the envelope theorem:*π πH = L = Q
*t+1 t π t t+1y - y = H y ,π ,t *
π t t+1= H y ,π ,t t+1 t t ty - y = Q y ,z ,t
![Page 25: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/25.jpg)
25
The Maximum Principle:
. .s t
t+1 t t t
t t
y - y = Q y ,z ,t
G y ,z ,t 0 max
T
t tt =0t t
y ,zF y ,z ,t
. .
For each : maximizes
the Hamiltonian s t t
t t t+1 t t
t z
H y ,z ,π ,t G y ,z
.
,t
1
0
t t t+1 t t t+1 t tH y ,z ,π ,t = F y ,z ,t π Q y ,z ,t
*t+1 t y t t+1 π - π = -H y ,π2. ,t
*t+1 t π t t+1 y -3. y = H y , π ,t
max s.t. *t t+1 t t t+1 t t
tzH y ,π ,t = H y ,z ,π ,t G y ,z ,t 0
![Page 26: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/26.jpg)
26
The Maximum Principle:
. .s t
t+1 t t t
t t
y - y = Q y ,z ,t
G y ,z ,t 0 max
T
t tt =0t t
y ,zF y ,z ,t
. .
For each : maximizes
the Hamiltonian s t t
t t t+1 t t
t z
H y ,z ,π ,t G y ,z
.
,t
1
0
t t t+1 t t t+1 t tH y ,z ,π ,t = F y ,z ,t π Q y ,z ,t
*t+1 t y t t+1 π - π = -H y ,π2. ,t
*t+1 t π t t+1 y -3. y = H y , π ,t
max s.t. *t t+1 t t t+1 t t
tzH y ,π ,t = H y ,z ,π ,t G y ,z ,t 0 t t t+1 t t t+1 t tH y ,z ,π ,t = F y ,z ,t π Q y ,z ,t
![Page 27: 1. 2 No lecture on Wed February 8th Thursday 9 th Feb Friday 27 th Jan Friday 10 th Feb Thursday 14:00 - 17:00 Friday 16:00 – 19:00 HS N.](https://reader036.fdocuments.us/reader036/viewer/2022062511/551c04bc550346a34f8b4dd3/html5/thumbnails/27.jpg)
27
The Maximum Principle:
. .s t
t+1 t t t
t t
y - y = Q y ,z ,t
G y ,z ,t 0 max
T
t tt =0t t
y ,zF y ,z ,t
. .
For each : maximizes
the Hamiltonian s t t
t t t+1 t t
t z
H y ,z ,π ,t G y ,z
.
,t
1
0
*t+1 t y t t+1 π - π = -H y ,π2. ,t
*t+1 t π t t+1 y -3. y = H y , π ,t
t t t+1 t t t+1 t tH y ,z ,π ,t = F y ,z ,t π Q y ,z ,t t t t+1 t t t tH y ,z ,π ,t = F y ,z ,t Q yt+1 ,z ,tπ