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Theorem Example Distances
Constrained Multivariable Optimization:Lagrange Multipliers
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Introduction
1. Finding extrema of functions of several variables onsurfaces by direct computation would be hard.
2. Lagrange multipliers reduce this task to a set of equations.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Introduction1. Finding extrema of functions of several variables on
surfaces by direct computation would be hard.
2. Lagrange multipliers reduce this task to a set of equations.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Introduction1. Finding extrema of functions of several variables on
surfaces by direct computation would be hard.2. Lagrange multipliers reduce this task to a set of equations.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem.
Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number.
If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k
, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f
10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f
10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10
20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10
20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20
30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20
30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30
40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30
40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40
50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40
50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
b
minimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
b
maximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .
The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k.
As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Theorem. Let f and g be two functions of equally manyvariables and let k be a number. If ~m is a local extremum of f onthe level (hyper)surface g(~r) = k, then there is a number λ suchthat ~∇f (~m) = λ~∇g(~m).
Explanation.
g = k
contours of f 10 20 30 40 50
bminimum
bmaximum
The gradient of f is perpendic-ular to the level surfaces of f .The gradient of g is perpen-dicular to the constraint sur-face g = k. As long as thesetwo gradients are not parallel,we can move inside the con-straint surface and increase ordecrease the function.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example.
Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
f (x,y) = 2x+3yg(x,y) = 5x2 +2y2 = 1
~∇f (x,y) = λ~∇g(x,y)[23
]= λ
[10x
4y
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
f (x,y) = 2x+3yg(x,y) = 5x2 +2y2 = 1
~∇f (x,y) = λ~∇g(x,y)[23
]= λ
[10x
4y
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
f (x,y) = 2x+3y
g(x,y) = 5x2 +2y2 = 1~∇f (x,y) = λ~∇g(x,y)[
23
]= λ
[10x
4y
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
f (x,y) = 2x+3yg(x,y) = 5x2 +2y2
= 1~∇f (x,y) = λ~∇g(x,y)[
23
]= λ
[10x
4y
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
f (x,y) = 2x+3yg(x,y) = 5x2 +2y2 = 1
~∇f (x,y) = λ~∇g(x,y)[23
]= λ
[10x
4y
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
f (x,y) = 2x+3yg(x,y) = 5x2 +2y2 = 1
~∇f (x,y) = λ~∇g(x,y)
[23
]= λ
[10x
4y
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
f (x,y) = 2x+3yg(x,y) = 5x2 +2y2 = 1
~∇f (x,y) = λ~∇g(x,y)[23
]
= λ
[10x
4y
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
f (x,y) = 2x+3yg(x,y) = 5x2 +2y2 = 1
~∇f (x,y) = λ~∇g(x,y)[23
]= λ
[10x
4y
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
f (x,y) = 2x+3yg(x,y) = 5x2 +2y2 = 1
~∇f (x,y) = λ~∇g(x,y)[23
]= λ
[10x
4y
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ10x = 2λ4y = 3 (???)
5x2 +2y2 = 1 !!!
Always remember the constraint.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ10x = 2
λ4y = 3 (???)5x2 +2y2 = 1 !!!
Always remember the constraint.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ10x = 2λ4y = 3
(???)5x2 +2y2 = 1 !!!
Always remember the constraint.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ10x = 2λ4y = 3 (???)
5x2 +2y2 = 1 !!!
Always remember the constraint.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ10x = 2λ4y = 3 (???)
5x2 +2y2 = 1
!!!
Always remember the constraint.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ10x = 2λ4y = 3 (???)
5x2 +2y2 = 1 !!!
Always remember the constraint.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ10x = 2λ4y = 3 (???)
5x2 +2y2 = 1 !!!
Always remember the constraint.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
x =1
5λ
y =3
4λ
5x2 +2y2 = 1
5(
15λ
)2
+2(
34λ
)2
= 1
15λ 2 +
98λ 2 = 1
5340λ 2 = 1 λ =±
√5340
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
x =1
5λ
y =3
4λ
5x2 +2y2 = 1
5(
15λ
)2
+2(
34λ
)2
= 1
15λ 2 +
98λ 2 = 1
5340λ 2 = 1 λ =±
√5340
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
x =1
5λ
y =3
4λ
5x2 +2y2 = 1
5(
15λ
)2
+2(
34λ
)2
= 1
15λ 2 +
98λ 2 = 1
5340λ 2 = 1 λ =±
√5340
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
x =1
5λ
y =3
4λ
5x2 +2y2 = 1
5(
15λ
)2
+2(
34λ
)2
= 1
15λ 2 +
98λ 2 = 1
5340λ 2 = 1 λ =±
√5340
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
x =1
5λ
y =3
4λ
5x2 +2y2 = 1
5(
15λ
)2
+2(
34λ
)2
= 1
15λ 2 +
98λ 2 = 1
5340λ 2 = 1 λ =±
√5340
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
x =1
5λ
y =3
4λ
5x2 +2y2 = 1
5(
15λ
)2
+2(
34λ
)2
= 1
15λ 2 +
98λ 2 = 1
5340λ 2 = 1 λ =±
√5340
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
x =1
5λ
y =3
4λ
5x2 +2y2 = 1
5(
15λ
)2
+2(
34λ
)2
= 1
15λ 2 +
98λ 2 = 1
5340λ 2 = 1
λ =±√
5340
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
x =1
5λ
y =3
4λ
5x2 +2y2 = 1
5(
15λ
)2
+2(
34λ
)2
= 1
15λ 2 +
98λ 2 = 1
5340λ 2 = 1 λ =±
√5340
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ =
√5340
x =1
5λ=
15
√4053
=
√8
265
y =3
4λ=
34
√4053
=
√45
106
f
(√8
265,
√45
106
)≈ 2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ =
√5340
x =1
5λ=
15
√4053
=
√8
265
y =3
4λ=
34
√4053
=
√45
106
f
(√8
265,
√45
106
)≈ 2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ =
√5340
x =1
5λ
=15
√4053
=
√8
265
y =3
4λ=
34
√4053
=
√45
106
f
(√8
265,
√45
106
)≈ 2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ =
√5340
x =1
5λ=
15
√4053
=
√8
265
y =3
4λ=
34
√4053
=
√45
106
f
(√8
265,
√45
106
)≈ 2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ =
√5340
x =1
5λ=
15
√4053
=
√8
265
y =3
4λ=
34
√4053
=
√45
106
f
(√8
265,
√45
106
)≈ 2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ =
√5340
x =1
5λ=
15
√4053
=
√8
265
y =3
4λ
=34
√4053
=
√45
106
f
(√8
265,
√45
106
)≈ 2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ =
√5340
x =1
5λ=
15
√4053
=
√8
265
y =3
4λ=
34
√4053
=
√45
106
f
(√8
265,
√45
106
)≈ 2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ =
√5340
x =1
5λ=
15
√4053
=
√8
265
y =3
4λ=
34
√4053
=
√45
106
f
(√8
265,
√45
106
)≈ 2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ =
√5340
x =1
5λ=
15
√4053
=
√8
265
y =3
4λ=
34
√4053
=
√45
106
f
(√8
265,
√45
106
)≈ 2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ=−1
5
√4053
=−√
8265
y =3
4λ=−3
4
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ=−1
5
√4053
=−√
8265
y =3
4λ=−3
4
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ
=−15
√4053
=−√
8265
y =3
4λ=−3
4
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ=−1
5
√4053
=−√
8265
y =3
4λ=−3
4
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ=−1
5
√4053
=−√
8265
y =3
4λ=−3
4
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ=−1
5
√4053
=−√
8265
y =3
4λ
=−34
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ=−1
5
√4053
=−√
8265
y =3
4λ=−3
4
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ=−1
5
√4053
=−√
8265
y =3
4λ=−3
4
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ=−1
5
√4053
=−√
8265
y =3
4λ=−3
4
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the extreme values of the functionf (x,y) = 2x+3y on the ellipse 5x2 +2y2 = 1.
λ = −√
5340
x =1
5λ=−1
5
√4053
=−√
8265
y =3
4λ=−3
4
√4053
=−√
45106
f
(−√
8265
,−√
45106
)≈ −2.3022
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example.
Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.The constraint is
g(x,y,z) := x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.The constraint is
g(x,y,z) := x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2))
=√(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.The constraint is
g(x,y,z) := x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√
(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.The constraint is
g(x,y,z) := x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√
(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.The constraint is
g(x,y,z) := x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√
(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance
, which is just as well.The constraint is
g(x,y,z) := x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√
(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.
The constraint is
g(x,y,z) := x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√
(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.The constraint is
g(x,y,z) := x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√
(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.The constraint is
g(x,y,z) :=
x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√
(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.The constraint is
g(x,y,z) := x2 + y2− z
= 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
d((x,y,z),(0,1,2)) =√
(x−0)2 +(y−1)2 +(z−2)2
=
√x2 +(y−1)2 +(z−2)2
Minimizing
f (x,y,z) := x2 +(y−1)2 +(z−2)2
gives the square of the minimum distance, which is just as well.The constraint is
g(x,y,z) := x2 + y2− z = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
f (x,y,z) = x2 +(y−1)2 +(z−2)2
g(x,y,z) = x2 + y2− z = 0~∇f (x,y,z) = λ~∇g(x,y,z) 2x
2(y−1)2(z−2)
= λ
2x2y−1
2x = λ2x
2(y−1) = λ2y2(z−2) = λ (−1)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
f (x,y,z) = x2 +(y−1)2 +(z−2)2
g(x,y,z) = x2 + y2− z = 0~∇f (x,y,z) = λ~∇g(x,y,z) 2x
2(y−1)2(z−2)
= λ
2x2y−1
2x = λ2x
2(y−1) = λ2y2(z−2) = λ (−1)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
f (x,y,z) = x2 +(y−1)2 +(z−2)2
g(x,y,z) = x2 + y2− z = 0
~∇f (x,y,z) = λ~∇g(x,y,z) 2x2(y−1)2(z−2)
= λ
2x2y−1
2x = λ2x
2(y−1) = λ2y2(z−2) = λ (−1)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
f (x,y,z) = x2 +(y−1)2 +(z−2)2
g(x,y,z) = x2 + y2− z = 0~∇f (x,y,z) = λ~∇g(x,y,z)
2x2(y−1)2(z−2)
= λ
2x2y−1
2x = λ2x
2(y−1) = λ2y2(z−2) = λ (−1)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
f (x,y,z) = x2 +(y−1)2 +(z−2)2
g(x,y,z) = x2 + y2− z = 0~∇f (x,y,z) = λ~∇g(x,y,z) 2x
2(y−1)2(z−2)
= λ
2x2y−1
2x = λ2x
2(y−1) = λ2y2(z−2) = λ (−1)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
f (x,y,z) = x2 +(y−1)2 +(z−2)2
g(x,y,z) = x2 + y2− z = 0~∇f (x,y,z) = λ~∇g(x,y,z) 2x
2(y−1)2(z−2)
= λ
2x2y−1
2x = λ2x2(y−1) = λ2y2(z−2) = λ (−1)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
f (x,y,z) = x2 +(y−1)2 +(z−2)2
g(x,y,z) = x2 + y2− z = 0~∇f (x,y,z) = λ~∇g(x,y,z) 2x
2(y−1)2(z−2)
= λ
2x2y−1
2x = λ2x
2(y−1) = λ2y2(z−2) = λ (−1)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
f (x,y,z) = x2 +(y−1)2 +(z−2)2
g(x,y,z) = x2 + y2− z = 0~∇f (x,y,z) = λ~∇g(x,y,z) 2x
2(y−1)2(z−2)
= λ
2x2y−1
2x = λ2x
2(y−1) = λ2y
2(z−2) = λ (−1)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
f (x,y,z) = x2 +(y−1)2 +(z−2)2
g(x,y,z) = x2 + y2− z = 0~∇f (x,y,z) = λ~∇g(x,y,z) 2x
2(y−1)2(z−2)
= λ
2x2y−1
2x = λ2x
2(y−1) = λ2y2(z−2) = λ (−1)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λxy−1 = λy
2(z−2) = −λ
x2 + y2− z = 0
λ = 1 ⇒ y−1 = y (not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λx
y−1 = λy2(z−2) = −λ
x2 + y2− z = 0
λ = 1 ⇒ y−1 = y (not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λxy−1 = λy
2(z−2) = −λ
x2 + y2− z = 0
λ = 1 ⇒ y−1 = y (not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λxy−1 = λy
2(z−2) = −λ
x2 + y2− z = 0
λ = 1 ⇒ y−1 = y (not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λxy−1 = λy
2(z−2) = −λ
x2 + y2− z = 0
λ = 1 ⇒ y−1 = y (not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λx ⇒ x = 0 or λ = 1y−1 = λy
2(z−2) = −λ
x2 + y2− z = 0
λ = 1 ⇒ y−1 = y (not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λx ⇒ x = 0 or λ = 1y−1 = λy
2(z−2) = −λ
x2 + y2− z = 0
λ = 1
⇒ y−1 = y (not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λx ⇒ x = 0 or λ = 1y−1 = λy
2(z−2) = −λ
x2 + y2− z = 0
λ = 1 ⇒ y−1 = y
(not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λx ⇒ x = 0 or λ = 1y−1 = λy
2(z−2) = −λ
x2 + y2− z = 0
λ = 1 ⇒ y−1 = y (not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = λx ⇒ x = 0 or λ = 1y−1 = λy
2(z−2) = −λ
x2 + y2− z = 0
λ = 1 ⇒ y−1 = y (not possible)
Hence x = 0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
y−1 = λy2(z−2) = −λ
y2− z = 0z = y2
λ = −2(
y2−2)
y−1 = −2(
y2−2)
y =−2y3 +4y
2y3−3y−1 = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
y−1 = λy
2(z−2) = −λ
y2− z = 0z = y2
λ = −2(
y2−2)
y−1 = −2(
y2−2)
y =−2y3 +4y
2y3−3y−1 = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
y−1 = λy2(z−2) = −λ
y2− z = 0z = y2
λ = −2(
y2−2)
y−1 = −2(
y2−2)
y =−2y3 +4y
2y3−3y−1 = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
y−1 = λy2(z−2) = −λ
y2− z = 0
z = y2
λ = −2(
y2−2)
y−1 = −2(
y2−2)
y =−2y3 +4y
2y3−3y−1 = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
y−1 = λy2(z−2) = −λ
y2− z = 0z = y2
λ = −2(
y2−2)
y−1 = −2(
y2−2)
y =−2y3 +4y
2y3−3y−1 = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
y−1 = λy2(z−2) = −λ
y2− z = 0z = y2
λ = −2(
y2−2)
y−1 = −2(
y2−2)
y =−2y3 +4y
2y3−3y−1 = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
y−1 = λy2(z−2) = −λ
y2− z = 0z = y2
λ = −2(
y2−2)
y−1 = −2(
y2−2)
y
=−2y3 +4y
2y3−3y−1 = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
y−1 = λy2(z−2) = −λ
y2− z = 0z = y2
λ = −2(
y2−2)
y−1 = −2(
y2−2)
y =−2y3 +4y
2y3−3y−1 = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
y−1 = λy2(z−2) = −λ
y2− z = 0z = y2
λ = −2(
y2−2)
y−1 = −2(
y2−2)
y =−2y3 +4y
2y3−3y−1 = 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
2y3−3y−1 = 0
(y+1)(
2y2−2y−1)
= 0
y1 =−1 y2,3 =2±√
4+84
=1±√
32
≈ 1.3660, −0.3660
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
2y3−3y−1 = 0
(y+1)(
2y2−2y−1)
= 0
y1 =−1 y2,3 =2±√
4+84
=1±√
32
≈ 1.3660, −0.3660
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
2y3−3y−1 = 0
(y+1)(
2y2−2y−1)
= 0
y1 =−1 y2,3 =2±√
4+84
=1±√
32
≈ 1.3660, −0.3660
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
2y3−3y−1 = 0
(y+1)(
2y2−2y−1)
= 0
y1 =−1
y2,3 =2±√
4+84
=1±√
32
≈ 1.3660, −0.3660
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
2y3−3y−1 = 0
(y+1)(
2y2−2y−1)
= 0
y1 =−1 y2,3 =2±√
4+84
=1±√
32
≈ 1.3660, −0.3660
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
2y3−3y−1 = 0
(y+1)(
2y2−2y−1)
= 0
y1 =−1 y2,3 =2±√
4+84
=1±√
32
≈ 1.3660, −0.3660
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
2y3−3y−1 = 0
(y+1)(
2y2−2y−1)
= 0
y1 =−1 y2,3 =2±√
4+84
=1±√
32
≈ 1.3660
, −0.3660
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
2y3−3y−1 = 0
(y+1)(
2y2−2y−1)
= 0
y1 =−1 y2,3 =2±√
4+84
=1±√
32
≈ 1.3660, −0.3660
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0
, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2
y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1
z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1
d((0,−1,1),(0,1,2)) =√
5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2))
=√
5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5
≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
x = 0, z = y2 y1 = −1 z1 = 1d((0,−1,1),(0,1,2)) =
√5≈ 2.2361
y2 =1−√
32
z2 =
(1−√
32
)2
d
0,1−√
32
,
(1−√
32
)2 ,(0,1,2)
≈ 2.3126
y3 =1+√
32
z3 =
(1+√
32
)2
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
So the shortest distance is
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
So the shortest distance is
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
So the shortest distance is
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
Theorem Example Distances
Example. Find the shortest distance between the point (0,1,2)and the paraboloid z = x2 + y2.
So the shortest distance is
d
0,1+√
32
,
(1+√
32
)2 ,(0,1,2)
≈ 0.3897.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Constrained Multivariable Optimization: Lagrange Multipliers
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