Post on 02-Jun-2018
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Lagrange MultipliersByDr. Julia Arnold
Professor of MathematicsTidewater Community College, Norfolk Campus,
Norfolk, VAWith Assistance from a CPDP Grant
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In this lesson you will
•Understand the method of Lagrange Multipliers
•Use Lagrange Multipliers to solve constrainedoptimization problems
•Use the method of Lagrange multipliers with twoconstraints
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Objective 1
Understand the method of
Lagrange Multipliers
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Fig.13.77
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2 2
2 2: ( , ) 1
3 4
x yConstraint g x y : ( , ) 4Objective function f x y xy
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Recall
Fig. 13.48
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Objective 2
Use Lagrange Multipliers to solve
constrained optimization problems
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Basically, Lagrange Multipliers are another way to think aboutsolving optimization problems with constraints. Lets look at aproblem we can solve either way.
Problem: Suppose I have 50 feet of fencing for arectangular shaped garden and want to enclosethe maximum area.
The usual way we would solve this problem is to look at apicture and put in variables.
x
y
Next we write the constraintequation 2x + 2y = 50
Last we determine we want tomaximize the Area of theGarden whose formula is
A = xy
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Problem: Suppose I have 50 feet of fencing for arectangular shaped garden and want to enclosethe maximum area.
x
y
2x + 2y = 50 Constraint
A = xy Objective
We want to get the Area in terms of just x or y so we use the constraint to eliminate one orthe other.
2y=50-2x
Y = 25-x
A = x (25-x)=25x – x2
A’ = 25 – 2x
0= 25 – 2x
X = 12.5 and y = 12.5
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Now we will use the method of Lagrange Multipliers toSolve the same problem.
2x + 2y = 50 Constraint
A = xy Objective
f(x,y)=xy is the objective equation and g(x,y)=2x +2y is the constraintequation. ( , ) ; ( , )
( , ) 2; ( , ) 2
2
2
2 2 50
4 4 50
8 50
6.25
12.5
12.5
x y
x y
x x
y y
f x y y f x y x
g x y g x y
Set f g y
Set f g x
x y
Solve
x
y
simultaneously
Find
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Now let’s look at a problem that might be difficult tosolve the old way.
Problem 2: Find the maximum and minimum values for2 2
4 4 4 4
3 3
3
3 3 22
4 4
3 22
2 22 2
( , )
( , ) 1
2 ; 2
4 ; 4
2 4
12 4 2 4 2 (1 2 ) 02
1
12 4 2 (1 2 ) 02
1 1:2 2
x y
x y
f x y x y objective equation
g x y x y with the x y
Find f x f y
g x g y
x x
y y x x x x x or x
x y
y y y y y or y
Solve x y x y
constraint
4
4 4 4 4 4 4
4 4
4 4
0 1 1
1 1 2 1 0 1 11 1
2 2
1 1, max 2
2 2
(0, 1), ( 1, 0) min 1
If x then y y
x y x x x IIf y then x x
x and y
We could have the following which yield a value of
and which yields a value of
points
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The constraint is
the red graph.The blue graphand green graphare level curves ofthe paraboloid.
The blue graph isat 1 and the greengraph is at
which is the exactvalue for the max.
As the paraboloidextends in the zdirection it getswider than theconstraint.
Z=1
x
y
Z= 2
2
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Objective 3Use the method of Lagrange
multipliers with two constraints
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Suppose we want to find the max and min values of f(x,y,z)subject to two constraints of the form g(x,y,z)=p and
h(x,y,z)= q. Geometrically, this means that we are looking forthe extreme values of f when (x,y,z) is restricted to lie onthe curve of intersection of the level surfaces g and h. It canbe shown that if an extreme value occurs at ,then the gradient vector is in the planedetermined by and .
We assume these gradient vectors are not 0 or parallel andthus there are numbers (Lagrange multipliers)such that
, ,o o o x y z ( , , )o o o f x y z
( , , )o o o g x y z ( , , )o o oh x y z
and
( , , ) ( , , ) ( , , )o o o o o o o o o f x y z g x y z h x y z
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Problem: Maximize f(x,y,z) = xyz subject to the twoconstraints 2 2 5 2 0 x z and x y
2 2
2 2
2 2
( , , )
( , , ) 2 0 2
( , , ) 2 1 2 0
5 2 0
2
0 2
2 0
5 2 0
x y z
x y z
x y z
f x y z xyz f yz f xz f xy
g x y z x z g x g g z
h x y z x y h h h
x z and x y
Solve
yz x
xz
xy z
x z and x y
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Solved eq 2 for mu
Solved eq 3 for lambda
Substituted into eq 1 for mu
and lambda.
2 2
2
0 2
2 0
5 2 0
22
22
22 2
Solve
yz x
xz
xy z
x z and x y
xz xz
xy xy z
z
xy xz yz x
z
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Continued
2 2 2
2 2 2 2
2 2 2
2 2 2
2 3
3 3
3
2
2
22
22
22 2
2 2
2 02
5 5
2 5 2 52 2
5 5
2 5
10 2 010 3 0
(10 3 ) 0
10 100,
3 3
xz xz
xy xy z
z
xy xz yz x
z
yz x y xz
xSub y from x y
Sub z x from x z x x
x x x x
x x x x x x
x x x
x x x x x
x x
x or x x
2 2
2
2
2
5
10 10
0, 3 3
0, 0, 5
10 1 10, ,
3 2 3
10 5 55 ,
3 3 3
10 1 10 5, ,
3 2 3 3
10 1 10 5, ,
3 2 3 3
10 1 10 5 5 15, ,
3 2 3 3 9
10 1 10 5 5 15, ,
3 2 3 3 9
x y
z x
x or x x
for x y z
for x y
z z
f
f
f
points (0,0, 5),
(0,0, 5) = 0
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I found this very interesting website written bya student who is sharing his talents witheveryone. The topic is Lagrange Multipliers.
Check it out.http://www.youtube.com/watch?v=ry9cgNx1QV8
His method is slightly different from the way yourtext presented it. Are they equivalent?
For comments on this presentation you may email the author Dr.
Julia Arnold at jarnold@tcc.edu.