1. 1 Mathe III Lecture 9 Mathe III Lecture 9

2. 2 Envelope Theorem

3. 3 First, consider the case where there are no costraints:

4. 4 Envelope Theorem (with no onstraints)

5. 5 Envelope Theorem At the maximum point: (with constraints)

6. 6 Envelope Theorem (with constraints)

7. 7 Envelope Theorem (with constraints) Lagrange:

8. 8 Envelope Theorem (with constraints)

9. 9 Non Linear Programming A simple case:

10. 10 x y + ++

11. 11 x y + ++ The max is a small change in c has no effect on the max.

12. 12 x y If the situation is : the max. is:

13. 13 Non Linear Programming a general treatment

14. 14 Non Linear Programming a general treatment The solution is a stationary point of the Lagrangian: clearly an increase in c i increases the maximum. Hence:

15. 15 Non Linear Programminga general treatment

16. 16 Non Linear Programming an example

17. 17 Non Linear Programming an example

18. 18 Non Linear Programming an example

19. 19 Non Linear Programming an example

20. 20 Non Linear Programming an example

21. 21 Non Linear Programming an example

22. 22 Non Linear Programming an example