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# Optimization with Equality Constraints - PowerPoint PPT Presentation

Optimization with Equality Constraints. Optimum Values and Extreme Values of a Function of two Variables. Local maximum. Local minimum. Extreme Values of a Function of two Variables. Saddle point. First Order Conditions ( The necessary conditions).

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### Optimization with Equality Constraints

Optimum Values and Extreme Values of a Function of two Variables

Local maximum

Local minimum

First Order Conditions(The necessary conditions)

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

Z=f(x ,y )

Finding the Stationary Values solutions of the following system:

• Z=f(x,y)=x2+y2

• Z=f(x,y)=x2-y2

• Z=f(x,y)=xy

The Hessian Matrix

H(x0,y0)>0 fxx >0 minimum

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

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

subject to constraints:

First Order Conditions(The necessary conditions)

For instance minimize the objective function

Subject to the constraint:

Direct methode

We can combine the constraint with the objective function:

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

Direct methode

Lagrangemultiplier

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

Lagrangemultiplier

Bordered Hessian Matrix of the Second Order derivative is given by

The point is a minimum

The point is a maximum

Lagrangemultiplier

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

with constraints

First OrderConditionsNecessaryCondition

We build a Lagrangian function :

Finding the Stationary Values:

Second Order ConditionsSufficient Conditions

• Second order conditions:

• We must check the sign of a Bordered Hessian:

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

NumericalExamples

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