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2 Local maximum Local minimum
3 Saddle point
4 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)=x 2 +y 2 2) Z=f(x,y)=x 2 -y 2 3) Z=f(x,y)=xy 5
6 The Hessian Matrix H(x 0,y 0 )>0 f xx >0 minimum H(x 0,y 0 )>0 f x x <0 maximum H(x 0,y 0 )<0 saddle
The method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function: subject to constraints: 7
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10 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) 11
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We introduce a new variable ( λ ) called a Lagrange multiplier, and study the Lagrange function: 13
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16 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 17
We build a Lagrangian function : Finding the Stationary Values: 18
Second order conditions: We must check the sign of a Bordered Hessian: 19
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 20
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Case n=3 e m=1 : 22
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n=3 e m=2 the matrix of the second order derivate is given by: 24
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