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