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Simplex

Simplex. “walk on the vertices of the feasible region”. v = current vertex if  neighbor v’ of v with better objective then move to v’. Simplex. “walk on the vertices of the feasible region”. vertex = feasible point defined by a collection of d inequalities.

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Simplex

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  1. Simplex “walk on the vertices of the feasible region” v = current vertex if  neighbor v’ of v with better objective then move to v’

  2. Simplex “walk on the vertices of the feasible region” vertex = feasible point defined by a collection of d inequalities neighbors = vertices sharing d-1 of the inequalities

  3. v = current vertex if  neighbor v’ of v with better objective then move to v’ Simplex max cT x Ax  b x  0 assume v = (0,...,0)T  i such that ci > 0 iff v is not optimal

  4. v = current vertex if  neighbor v’ of v with better objective then move to v’ Simplex max cT x Ax  b x  0 v’ = (0,..,xi,..,0)T Make xi as big as possible stopper: aj x = bj

  5. v = current vertex if  neighbor v’ of v with better objective then move to v’ Simplex max cT x Ax  b x  0 v’ = (0,..,xi,..,0)T Make xi as big as possible stopper: aj x = bj Substitute: xi’ = bj – aj x

  6. Simplex max 2x+y x+y  4 y-x  1 x-y  2 y  2 x  0 y  0 Is (x,y)=(0,0) optimal?

  7. Simplex max 2x+y x+y  4 y-x  1 x-y  2 y  2 x  0 y  0 Let’s increase y as much as we can.

  8. Simplex max 2x+y x+y  4 y-x  1 x-y  2 y  2 x  0 y  0 substitute z=1-(y-x)

  9. Simplex max 2x+y x+y  4 y-x  1 x-y  2 y  2 x  0 y  0 substitute z=1-(y-x) z  0 y  x-z+1

  10. Simplex max 2x+y x+y  4 y-x  1 x-y  2 y  2 x  0 y  0 max 3x-z+1 2x-z  3 z  0 z  1 x-z  1 x  0 z-x  1 y  x-z+1

  11. Simplex max 3x-z+1 2x-z  3 z  0 z  1 x-z  1 x  0 z-x  1 Is (x,z)=(0,0) optimal?

  12. Simplex max 3x-z+1 2x-z  3 z  0 z  1 x-z  1 x  0 z-x  1 Let’s increase x as much as we can.

  13. Simplex max 3x-z+1 2x-z  3 z  0 z  1 x-z  1 x  0 z-x  1 substitute w=1-(x-z)

  14. Simplex max 3x-z+1 2x-z  3 z  0 z  1 x-z  1 x  0 z-x  1 substitute w=1-(x-z) w  0 x  1+z-w

  15. Simplex max 3x-z+1 2x-z  3 z  0 z  1 x-z  1 x  0 z-x  1 max 2z-3w+4 z-2w  1 z  0 z  1 w  0 w-z  1 w  2 x  1+z-w

  16. Simplex max 2z-3w+4 z-2w  1 z  0 z  1 w  0 w-z  1 w  2 Is (z,w)=(0,0) optimal?

  17. Simplex max 2z-3w+4 z-2w  1 z  0 z  1 w  0 w-z  1 w  2 Let’s increase z as much as we can.

  18. Simplex max 2z-3w+4 z-2w  1 z  0 z  1 w  0 w-z  1 w  2 substitute u=1-(z-2w)

  19. Simplex max 2z-3w+4 z-2w  1 z  0 z  1 w  0 w-z  1 w  2 substitute u=1-(z-2w) u  0 z  1+2w-u

  20. Simplex max 2z-3w+4 z-2w  1 z  0 z  1 w  0 w-z  1 w  2 max w-2u+6 u  0 u-2w  1 2w-u  2 w  0 u-w  2 w  2 z  1+2w-u

  21. Simplex max w-2u+6 u  0 u-2w  1 2w-u  2 w  0 u-w  2 w  2 Is (u,w)=(0,0) optimal?

  22. Simplex max w-2u+6 u  0 u-2w  1 2w-u  2 w  0 u-w  2 w  2 Let’s increase w as much as we can.

  23. Simplex max w-2u+6 u  0 u-2w  1 2w-u  2 w  0 u-w  2 w  2 substitute v=2-(2w-u)

  24. Simplex max w-2u+6 u  0 u-2w  1 2w-u  2 w  0 u-w  2 w  2 substitute v=2-(2w-u) v  0 w  1+u/2-v/2

  25. Simplex max w-2u+6 u  0 u-2w  1 2w-u  2 w  0 u-w  2 w  2 max 7-3u/2-v/2 u  0 v  3 v  0 v-u  2 u+v  6 u-v  2 w1+u/2-v/2

  26. Simplex max 7-3u/2-v/2 u  0 v  3 v  0 v-u  2 u+v  6 u-v  2 Is (u,v)=(0,0) optimal?

  27. Simplex max 7-3u/2-v/2 u  0 v  3 v  0 v-u  2 u+v  6 u-v  2 7 YES Is (u,v)=(0,0) optimal?

  28. Simplex (u,v)=(0,0) w  1+u/2-v/2 = 1 z  1+2w-u = 3 x  1+z-w = 3 y  x-z+1 = 1 (x,y)=(3,1)

  29. Simplex (x,y)=(3,1) max 2x+y x+y  4 y-x  1 x-y  2 y  2 x  0 y  0 is an optimal solution

  30. Simplex – geometric view (x,y)=(3,1) max 2x+y x+y  4 y-x  1 x-y  2 y  2 x  0 y  0

  31. Getting the first point min 1T z A x + z = b x  0 z  0 min cT x Ax=b x  0 wlog b  0

  32. Points, lines point = (x,y) line = (x1,y1),(x2,y2) = 2 points

  33. Line as a point and a vector point = (x,y) x1+t (x2-x1),y1+t (y2-y1) line = (x1,y1),(x2-x1,y2-y1) = point and a vector

  34. Is point on a line? point = (x,y) x=x1+t (x2-x1) y=y1+t (y2-y1) line = (x1,y1),(x2,y2)

  35. ) ( x2-x1 x-x1 y2-y1 y-y1 det Is point on a line? point = (x,y) t (x2-x1)=x-x_1 t (y2-y1)=y-y_1 line = (x1,y1),(x2,y2)

  36. ) ( x2-x1 x-x1 y2-y1 y-y1 det Is point on a line? point = (x,y) is on line = (x1,y1),(x2,y2) if and only if = 0

  37. ) ( x2-x1 x-x1 y2-y1 y-y1 det Is point on a line? =0 for x on the line >0 <0

  38. Line segment x=x1+t (x2-x1) y=y1+t (y2-y1) t  [0,1] line segment = (x1,y1),(x2,y2)

  39. Do two line segments intersect? a1=(x1,y1), a2=(x2,y2) a3=(x3,y3), a4= (x4,y4) a3 L2 L1 a2 a4 a1 a1 and a2 on different sides of L2 a3 and a4 on different sides of L1 or endpoint of a segment lies on the other segment

  40. Many segments, do any 2 intersect? (a1,b1) (a2,b2) ... (an,bn) O(n2) algorithm

  41. Many segments, do any 2 intersect? O(n log n) algorithm assume no two points have the same x-coordinate no 3 segments intersect at one point

  42. Sweep algorithm

  43. Sweep algorithm sort points by the x-coordinate

  44. Sweep algorithm events: insert segment delete segment

  45. Sweep algorithm will find the left-most intersection point the lines are “neighbors on the sweep line”

  46. Sweep algorithm sort the endpoints by x-coord  p1,...,p2n T empty B-tree for i from 1 to 2n do if pi is the left point of a segment s INSERT s into T check if s intersects prev(s) or next(s) in T if pi is the right point of a segment s check if prev(s) interesects next(s) in T DELETE s from T

  47. Area of a simple polygon (x3,y3) (x2,y2) (x1,y1)

  48. Area of a simple polygon (x1,y1),...,(xn,yn)

  49. Area of a simple polygon (x1,y1),...,(xn,yn) (xn+1,yn+1)=(x1,y1) R=0 for i from 1 to n do R=R+(yi+1+yi)*(xi+1-xi) return |R|/2

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