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2Local maximum Local minimum

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Saddle point

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Given the problem of maximizing ( or minimizing) of the objective function:

Z=f(x ,y )Finding the Stationary Values solutions of the following system:

1) Z=f(x,y)=x2+y2

2) Z=f(x,y)=x2-y2

3) Z=f(x,y)=xy

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The Hessian Matrix

H(x0,y0)>0 fxx >0 minimum H(x0,y0)>0 fxx <0 maximum H(x0,y0)<0 saddle

The method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function:

subject to constraints:

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For instance minimize the objective function

Subject to the constraint:

We can combine the constraint with the objective function:

Minimum in P(1/2;1/2)

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We introduce a new variable (λ) called a Lagrange multiplier, and study the Lagrange function:

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The point is a minimum

The point is a maximum

Bordered Hessian Matrix of the Second Order derivative is given by

Given the problem of maximizing ( or minimizing) of the objective function

with constraints

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1, , nz f x x

1 1 1

1

, ,

, ,

n

m n m

g x x c

m n

g x x c

We build a Lagrangian function :

Finding the Stationary Values:

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1

, ( ) ( )m

i i ii

L f g c

x λ x x

0 1, ,

00 1, ,

i

j

Li n

xL

Lj m

Second order conditions:

We must check the sign of a Bordered Hessian:

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2 ,Lx 0 0x λ

im m

j

T

i

j

g

x

g

x

L

0

H

n=2 e m=1 the Bordered Hessian Matrix of the Second Order derivative is given by

Det>0 imply Maximum Det<0 imply Minimum

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1 1

1 2

2 21

21 1 1 2

2 21

22 2 1 2

0g g

x x

g L L

x x x x

g L L

x x x x

3 3 2

2

Max/Min ( , ) 3 4

. . ( , ) 2 1 0

f x y x y x y x y

s v g x y x y

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Case n=3 e m=1 :

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1 1 1

1

1

1

0 x y z

x xx xy xz

y yx yy yz

z zx zy zz

g g g

g L L L

g L L L

g L L L

2 2

2

Max/Min ( , , ) 2

. . ( , , ) 1 0

f x y z x y x z

s v g x y z x y z

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n=3 e m=2 the matrix of the second order derivate is given by:

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1 1 1

2 2 2

1 2

1 2

1 2

0 0

0 0x y z

x y z

x x xx xy xz

y y yx yy yz

z z zx zy zz

g g g

g g g

g g L L L

g g L L L

g g L L L

2 2

2 2 2

Max/Min ( , , ) log 1

. . ( , , ) 16 0

( , , ) 4 0

f x y z x y x z

s v g x y z x y z

h x y z x y z

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