1. Mathe III Lecture 8 Mathe III Lecture 8

2. 2 Constrained Maximization Lagrange Multipliers At a maximum point of the original problem the derivatives of the Lagrangian vanish (w.r.t. all variables).

3. 3 Constrained Maximization Lagrange Multipliers Intuition x y iso- f curves f(x,y) = K 5 6 20 5 20 assume +

4. 4 Constrained Maximization Lagrange Multipliers Intuition x y

5. 5 Constrained Maximization Lagrange Multipliers Intuition x y

6. 6 Constrained Maximization Lagrange Multipliers Intuition x y

7. 7 Constrained Maximization Lagrange Multipliers Intuition x y

8. 8 Constrained Maximization Lagrange Multipliers Intuition A stationary point of the Lagrangian

9. 9 Constrained Maximization Lagrange Multipliers Intuition A stationary point of the Lagrangian

10. 10 Constrained Maximization The general case

11. 11 Constrained Maximization The general case differentiating w.r.t. x s, s = m+1,…,n

12. 12 Constrained Maximization The general case

13. 13 Constrained Maximization The general case

14. 14 Constrained Maximization The general case

15. 15 Constrained Maximization The general case The derivatives w.r.t. x m+1,…..x n are zero at a max (min) point.

16. 16 Constrained Maximization The general case

17. 17 Constrained Maximization The general case But:

18. 18 Constrained Maximization The general case

19. 19 Constrained Maximization The general case We need to show this for s = 1,….m

20. 20 Constrained Maximization The general case

21. 21 Constrained Maximization The general case

22. 22 Constrained Maximization The general case We have shown that a solution of the original problem satisfies

23. 23 Constrained Maximization The general case i.e. a solution of the original problem is a stationary point of the Lagrangian :

24. 24 Constrained Maximization Interpretation of the multipliers

25. 25 Constrained Maximization Interpretation of the multipliers But:

26. 26 Constrained Maximization Interpretation of the multipliers ??

27. 27 Constrained Maximization Interpretation of the multipliers differentiate w.r.t. c i

28. 28 Constrained Maximization Interpretation of the multipliers