1 / 28

Mathe III Lecture 8

Mathe III Lecture 8. Constrained Maximization. Lagrange Multipliers. At a maximum point of the original problem. the derivatives of the Lagrangian vanish (w.r.t. all variables). Constrained Maximization. Lagrange Multipliers. Intuition. y. iso- f curves. f(x,y) = K. 5. 20. 6.

rae
Download Presentation

Mathe III Lecture 8

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mathe III Lecture 8

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

  4. Constrained Maximization Lagrange Multipliers Intuition y x

  5. Constrained Maximization Lagrange Multipliers Intuition y x

  6. Constrained Maximization Lagrange Multipliers Intuition y x

  7. Constrained Maximization Lagrange Multipliers Intuition y x

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

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

  10. Constrained Maximization The general case

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

  12. The general case Constrained Maximization

  13. The general case Constrained Maximization

  14. The general case Constrained Maximization

  15. The general case Constrained Maximization The derivatives w.r.t. xm+1,…..xnare zero at a max (min) point.

  16. The general case Constrained Maximization

  17. The general case Constrained Maximization But:

  18. The general case Constrained Maximization

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

  20. The general case Constrained Maximization

  21. The general case Constrained Maximization

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

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

  24. Constrained Maximization Interpretation of the multipliers

  25. Constrained Maximization Interpretation of the multipliers But:

  26. Constrained Maximization Interpretation of the multipliers ??

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

  28. Constrained Maximization Interpretation of the multipliers

More Related