1
Mathe III
Jan 10, 2016
Mathe III. Lecture 10. 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. Non Linear Programming. Kuhn – Tucker conditions. Non Linear Programming. Kuhn – Tucker conditions. Nonnegative variables. - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
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
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
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
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
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
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 ...... ......
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
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).
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
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
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
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
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
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
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
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
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
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:
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
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
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
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
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
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
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
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
Related Documents
