Optimization with equality constraints
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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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Optimum values and extreme values of a function of two variables
Optimum Values and Extreme Values of a Function of two Variables

Local maximum

Local minimum



First order conditions the necessary conditions
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
Examples

  • Z=f(x,y)=x2+y2

  • Z=f(x,y)=x2-y2

  • Z=f(x,y)=xy


Second order conditions
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 variables1
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 conditions1
First Order Conditions(The necessary conditions)


Extreme values of a function of two variables2
Extreme Values of a Function of two Variables

For instance minimize the objective function

Subject to the constraint:


Direct methode
Direct methode

We can combine the constraint with the objective function:

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


Direct methode1
Direct methode


Lagrange multiplier
Lagrangemultiplier

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


Lagrange multiplier1
Lagrangemultiplier



Second order conditions1
Second Order Conditions

Bordered Hessian Matrix of the Second Order derivative is given by

The point is a minimum

The point is a maximum


Lagrange multiplier2
Lagrangemultiplier

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

with constraints


First order conditions necessary condition
First OrderConditionsNecessaryCondition

We build a Lagrangian function :

Finding the Stationary Values:


Second order conditions sufficient conditions
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


Numerical examples
NumericalExamples



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



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