lagrange multipliers erasmus mobility program (24apr2012) pollack mihály engineering faculty (pmmk)...

Lagrange Multipliers

Erasmus Mobility Program (24Apr2012)Pollack Mihály Engineering Faculty (PMMK)

University of Pécs

João Miranda

[email protected]

http://tiny.cc/jlm_estg ; http://tiny.cc/jlm_ist

24-Apr-12 Lagrange Multipliers 2


• Introduction: maximization of 2-variable function subject to one restriction;

• Generalization: maximization of n-variable function subject to m restrictions

• Tables & Chairs problem.



Optima of multi-variable functions when submitted to several constraints.

• Problem:

– Obtain the maxima/minima of the function,

– But the variables x and y have to satisfy the equation:

),( yxfu

0),( yxg


• Assuming y(x) as implicit function of x, the derivative of the composite function u(x,y),

(necessary condition)

• Or,


dx

du

0

0.

dx

dy

y

f

x

f

x

f

dx

dy

y

f


• The derivation of the implicit function, y(x):

then,


yg

xg

dx

dy

.y

g

x

g

x

g

dx

dy

dx

dy

y

g. 0


• Conjugating the two zero equations:

where is known as Lagrange multiplier (1736-1813).

(http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange) • Re-allocating terms,


dx

dy

y

f

x

f.

x

g

x

f. 0..

dx

dy

y

g

y

f

dx

dy

y

g

x

g.. 0


• The multiplier is selected in conformity with,

and it is also necessary that:

(http://en.wikipedia.org/wiki/Lagrange_multiplier)


0.

y

g

y

f

0.

x

g

x

f


• Thus, in optimal points, the three following equations are simultaneously verified (necessary conditions to the existence

of constrained optima):

• Notice that they represent the three partial derivatives in (x, y, ) of the function:

Verify it!


yxgyxfyxL ,.,),,(

0.

x

g

x

f

0.

y

g

y

f

0),( yxg


• Procedure:1. Build the Lagrangean function, L(x,y,) ;

2. Set to zero the related first order partial derivatives;

3. Obtain the values (x, y, ) that are satisfying the system of equations.



• Problem (general):

– Obtain the maxima/minima of the function,

– But the n variables have to simultaneously satisfy the set of m (m<n) equations:


),...,,( 21 nxxxfu ),...,,( 21 nxxx

0),...,,(

(...)

0),...,,(

0),...,,(

21

2212

1211

mnm

n

n

bxxxg

bxxxg

bxxxg


• Procedure:1. Build the Lagrangean function, ;


),...,,(),...,,,,...,,( 212121 nmn xxxfxxxL

12111 ),...,,(. bxxxg n

),( λxL

22122 ),...,,(. bxxxg n

mnm bxxxg ),...,,(.

...

211


• (...) Procedure:2. Set to zero the n first order partial derivatives in

the n variables ,

and...


1x

f

mnmmnn

n

bxxxgbxxxgbxxxg

xxxfL

),...,,(.(...)),...,,(.),...,,(.

),...,,(),(

212212212111

21

λx

1

11. x

g

2

11

2

.x

g

x

f

0......

(...)

22

11

n

mm

nnn x

g

x

g

x

g

x

f

1

22.

x

g

0....1

x

gmm

0.....22

22

x

g

x

g mm

),...,,( 21 nxxx


• (...) Procedure:3. ... Notice that the first order partial derivatives in

order to the m multipliers is driving the m constraints,


),...,,( 21 m

),...,,( 211 nxxxg

0),...,,(

(...)

21 mnm bxxxg

0),...,,( 2212 bxxxg n

01 b

mnmmnn

n

bxxxgbxxxgbxxxg

xxxfL

),...,,(.(...)),...,,(.),...,,(.

),...,,(),(

212212212111

21

λx


• (...) Procedure:3. Obtain the values set that simultaneously

satisfy the group of (n + m) equations. That is,


),( λx

njx

g

x

f

x

L m

i j

ii

jj

,1,0)(

.)(),(

1

xxx λ

mibgL

iii

,1,0)(),(

xx λ


A furniture factory builds Tables (t) at a profit of 4 Euros per Table, and Chairs (c) at a profit of 3 Euros per Chair).

Suppose that only 8 short (s) pieces and 6 large (l) pieces are available for building purposes, what combination of Tables and Chairs

do you need to build to make the most profit?

.

If the availability of the short pieces is 8008 and the availability of the large pieces is 6007,

how many Tables and Chairs do you need to build to make the most profit?

Tables & Chairs (T&C)


Note: modeling is based in the proportionality, aditivity and divisibility between the produced quantities of t and c

and profit and utilization of l and s components.

.


,subject to

ProfitTotalMaximize

tables

large

produce to

used

components

tables

short

produce to

used

components

chairs

large

produce to

used

components

components

ofty Availabili

large

tables

short

produce to

used

components

components

oftyAvailabili

short


Note: modeling is based in the proportionality, aditivity and divisibility between the produced quantities of t and c

and profit and utilization of l and s components.


Prftmax

0,

,

ct

tosubject

600712 ct

800822 ct

ct 34


T&C: Lagrange Multipliers

Maximize the profit function, Luc(t,c), satisfying the conditions

concerning the availability of the l and s components..

• Problem:

– Obtain the maximum value of profit function,

– But the variables t and c shall satisfy the availability of l and s:

ctPrft 34

800822

600712

ct

ct


• Procedure:1. Build the Lagrangean function, L(t, c, 1, 2) ;

2. Set to zero the 4 first order partial derivatives;

3. Obtain the values (t, c, 1, 2) that are satisfying the system of equations.



• Procedure:1. Build the Lagrangean function, L(t, c, 1, 2) ;


ct 34

600712.1 ct

),(.),(.),(),,,( 2121 ctsctlctPrftctL

800822.2 ct


• (...) Procedure:2. Set to zero the 4 first order partial derivatives, and

notice that those related to the multipliers (1, 2) are driving the original constraints:

and...


0224 21

0

0

0

0

2

1

L

Lc

Lt

L

ct 34

600712.1 ct

),,,( 21 ctL

800822.2 ct

0213 21

0600712 ct

0800822 ct


• (...) Procedure:3. Obtain the values (t, c, 1, 2) that are

simultaneously satisfying the 4 equations:

• Re-allocating the terms of the system of equations,


422 21 321 21

600712 ct

800822 ct

0224 21

0213 21

0600712 ct

0800822 ct




• Applying the Cramer’s Rule to the 1.st subsystem,


21

22

23

24

1

321

422

21

21

2.12.2

2.32.41

2.12.2

1.43.22

12

2

24

681

12

2

24

462

21

22

31

42

2




• Applying the Cramer’s Rule to the 2.nd subsystem,


22

12

28008

16007

t

800822

600712

ct

ct

1.22.2

1).8008(2).6007(t

1.22.2

)6007.(2)8008.(2

c

20032

4006

24

800812014t

20012

4002

24

1201416016

c

22

12

80082

60072

c



simultaneously satisfying the 4 equations :

• Then,

Optimal Solution!


2001

2003

1

1

2

1

c

t

Edit LINDO:max 4t + 3csubject tol1) 2t + 1c <= 6007l2) 2t + 2c <= 8008END


Lagrange Multipliers (synthesis)

• Introduction: maximization of 2-variable function subject to one restriction;

• Generalization: maximization of n-variable function subject to m restrictions

• Tables & Chairs problem.

lagrange multipliers erasmus mobility program (24apr2012) pollack mihály engineering faculty (pmmk)...

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