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Optimization with Equality ConstraintsPowerPoint Presentation

Optimization with Equality Constraints

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Extreme Values of a Function of two Variables

Saddle point

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:

Examples

- Z=f(x,y)=x2+y2
- Z=f(x,y)=x2-y2
- Z=f(x,y)=xy

Second Order Conditions

The Hessian Matrix

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

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

H(x0,y0)<0 saddle

Extreme Values of a Function of two Variables

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)

Extreme Values of a Function of two Variables

For instance minimize the objective function

Subject to the constraint:

Direct methode

Lagrangemultiplier

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

Lagrangemultiplier

Second Order Conditions

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

Case n=3 e m=1 :

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

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