1 / 30

OR II GSLM 52800

OR II GSLM 52800. Outline. separable programming quadratic programming. Separable Programs. a separable NLP if f and all g j are separable functions 0  x i   i , a finite number. Idea of Separable Program. min f (x), s . t . g j (x)  0 for j = 1, …, m .

rich
Download Presentation

OR II GSLM 52800

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. OR IIGSLM 52800 1

  2. Outline • separable programming • quadratic programming 2

  3. Separable Programs • a separable NLP if • f and all gjare separable functions • 0  xi i, a finite number 3

  4. Idea of Separable Program • min f(x), • s.t. • gj(x)  0 for j = 1, …, m. • hard NLP but simple LP problems • approximating a separable NL program by a LP • a non-linear function by a piecewise linear one 4

  5. A Fact About Convex Functions • f: a convex function • for any  > 0, possible to find a sequence of piecewise linear convex functions fnsuch that |f  fn|   5

  6. Example 6.1 • a separable program 6

  7. 20.25 C 16 B 9 4 A 1 O 3 4 1 2 Example 6.1 • approximating by a piecewise linear function • two representations, -form and -form 7

  8. 20.25 C 16 B 9 4 A 1 O 3 4 1 2 Form • a piecewise linear function with (segment) break points • any point = the convex combination of the two break points of the linear segment • i ( 0) = the weight of break point i 8

  9. Example 6.1 • the program becomes the last but one type of constraints is non-linear 9

  10. Fact • nonlinear constraint: at most two adjacent itaking non-zero values • possible to have only one i = 1 • for convex f and gj: no need to have the non-linear constraint • non-optimal to have more than two non-zero i, or two i not adjacent 10

  11. Fact • non-optimal to have more than two non-zero i, or two i not adjacent • e.g., f being an objective function • any convex combination between two non-adjacent break points being above the piecewise non-linear function • similarly, the point for three or more non-zero I’s lying above the piecewise non-linear function • think about A = 0.3, B = 0.4, and C = 0.3 20.25 C 16 B 9 4 A 1 O 11 3 4 1 2

  12. Fact • non-optimal to have more than two non-zero i, or two i not adjacent • e.g., gj being a constraint • gj(0.3A+0.7B)  gj(0.3A+0.7C)  bj the feasible set of {0.3A+0.7B} is larger than that by {0.3A+0.7C}  the solution from {0.3A+0.7C} cannot be minimum • similar argument for three or more non-zero i’s lying above the piecewise non-linear function C B 12 A O

  13. Example 6.1 • the program becomes a linear program 13

  14. Example 6.2: Non-Convex Problem • min f(x), • s.t. 1 x 3. • approximating f(x) by a piecewise linear function • y = 0 + 10A + 6B • x = 0 + 2A + 3B 14

  15. Example 6.2: Non-Convex Problem • adding slack variable s, surplus variable u, and artificial variable a1 and a2: 15

  16. Example 6.2: Non-Convex Problem 16

  17. Example 6.2: Non-Convex Problem 17

  18. Example 6.2: Non-Convex Problem • most negative 0 • B in basis  only A qualified to enter, not O 18

  19. Example 6.2: Non-Convex Problem 19

  20. Example 6.2: Non-Convex Problem 20

  21. Example 6.2: Non-Convex Problem 21

  22. 20.25 C 16 B 9 4 A 1 O 3 4 1 2 Form • again, the last constraint is unnecessary for a convex program 22

  23. Quadratic Programming 23

  24. Quadratic Objective Function & Linear Constraints • Langrangian function 24

  25. KKT Conditions • positive definite Q • a convex program • a unique global minimum • the KKT sufficient • otherwise, KKT necessary 25

  26. KKT Conditions • cT + xTQ + TA  0  Qx + A y = c • Ax  b  0  Ax + v = b • xT(c + Qx + A) = 0  xTy = 0 • T(Ax  b) = 0  Tv = 0 • x  0,  0, y  0, v  0 • solving the set of equations  phase-1 of a linear program 26

  27. Example 7.1(Example 10.14 of JB) 27

  28. Example 7.1(Example 10.14 of JB) • KKT conditions • 2x1 + 1 + 2y1 = 8, • 8x2+ 1y2 = 16, • x1+ x2 + v1 = 5, • x1 + v2 = 3. • x1y1 = x2y2 = 1v1 = 2v2 = 0 • x1, y1, x2, y2, 1, v1, 2, v2 0 28

  29. Example 7.1(Example 10.14 of JB) 29

  30. Example 7.1(Example 10.14 of JB) 30

More Related