Presentation is loading. Please wait.

Presentation is loading. Please wait.

Mathe III Lecture 8 Mathe III Lecture 8. 2 Constrained Maximization Lagrange Multipliers At a maximum point of the original problem the derivatives of.

Similar presentations


Presentation on theme: "Mathe III Lecture 8 Mathe III Lecture 8. 2 Constrained Maximization Lagrange Multipliers At a maximum point of the original problem the derivatives of."— Presentation transcript:

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 The general case

10 10 Constrained Maximization The general case differentiating w.r.t. x s, s = m+1,…,n

11 11 Constrained Maximization The general case

12 12 Constrained Maximization The general case

13 13 Constrained Maximization The general case

14 14 Constrained Maximization The general case The derivatives w.r.t. x m+1,…..x n are zero at a max (min) point.

15 15 Constrained Maximization The general case

16 16 Constrained Maximization The general case But:

17 17 Constrained Maximization The general case

18 18 Constrained Maximization The general case We need to show this for s = 1,….m

19 19 Constrained Maximization The general case

20 20 Constrained Maximization The general case

21 21 Constrained Maximization The general case define:

22 22 Constrained Maximization Interpretation of the multipliers

23 23 Constrained Maximization Interpretation of the multipliers But:

24 24 Constrained Maximization Interpretation of the multipliers

25 25 Constrained Maximization Interpretation of the multipliers 9


Download ppt "Mathe III Lecture 8 Mathe III Lecture 8. 2 Constrained Maximization Lagrange Multipliers At a maximum point of the original problem the derivatives of."

Similar presentations


Ads by Google