Optimization with Equality Constraints

1. Optimization with Equality Constraints

2. Optimum Values and Extreme Values of a Function of two Variables Local maximum Local minimum

3. Extreme Values of a Function of two Variables Saddle point

4. 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:

5. Examples • Z=f(x,y)=x2+y2 • Z=f(x,y)=x2-y2 • Z=f(x,y)=xy

6. 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

7. 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:

8. First Order Conditions(The necessary conditions)

10. Extreme Values of a Function of two Variables For instance minimize the objective function Subject to the constraint:

11. Direct methode We can combine the constraint with the objective function: Minimum in P(1/2;1/2)

12. Direct methode

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

14. Lagrangemultiplier

15. First Order Conditions

16. Second Order Conditions Bordered Hessian Matrix of the Second Order derivative is given by The point is a minimum The point is a maximum

17. Lagrangemultiplier Given the problem of maximizing ( or minimizing) of the objective function with constraints

18. First OrderConditionsNecessaryCondition We build a Lagrangian function : Finding the Stationary Values:

19. Second Order ConditionsSufficient Conditions • Second order conditions: • We must check the sign of a Bordered Hessian:

20. 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

21. NumericalExamples

22. Case n=3 e m=1  :

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

25. Example