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Linear programming

Linear programming. maximize x 1 + x 2 x 1 + 3x 2  3 3x 1 + x 2  5 x 1  0 x 2  0. Linear programming. maximize x 1 + x 2 x 1 + 3x 2  3 3x 1 + x 2  5 x 1  0 x 2  0. x 1. x 2. Linear programming. maximize x 1 + x 2 x 1 + 3x 2  3 3x 1 + x 2  5

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Linear programming

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  1. Linear programming maximize x1 + x2 x1 + 3x2 3 3x1 + x2 5 x1  0 x2  0

  2. Linear programming maximize x1 + x2 x1 + 3x2 3 3x1 + x2 5 x1  0 x2  0 x1 x2

  3. Linear programming maximize x1 + x2 x1 + 3x2 3 3x1 + x2 5 x1  0 x2  0 x1 x2

  4. Linear programming maximize x1 + x2 x1 + 3x2 3 3x1 + x2 5 x1  0 x2  0 x1 feasible solutions x2

  5. Linear programming maximize x1 + x2 x1 + 3x2 3 3x1 + x2 5 x1  0 x2  0 x1 optimal solution x1=1/2, x2=3/2 x2

  6. Can you prove it is optimal ? maximize x1 + x2 x1 + 3x2 3 3x1 + x2 5 x1  0 x2  0 x1 optimal solution x1=1/2, x2=3/2 x2

  7. Can you prove it is optimal ? maximize x1 + x2 x1 + 3x2 3 3x1 + x2 5 4x1 + 4x2  8 x1 optimal solution x1=1/2, x2=3/2 x2

  8. Can you prove it is optimal ? maximize x1 + x2 x1 + 3x2 3 3x1 + x2 5 x1+x2  2 x1 optimal solution x1=1/2, x2=3/2 x2

  9. Another linear program maximize x1 + x2 x1 + 2x2 3 4x1 + x2 5 x1  0 x2  0

  10. Another linear program maximize x1 + x2 x1 + 2x2 3 4x1 + x2 5 x1  0 x2  0 x1=1, x2=1, optimal ?

  11. Another linear program maximize x1 + x2 x1 + 2x2 3 *3 4x1 + x2 5 *1 x1  0 x2  0 7x1 + 7x2 14 x1=1, x2=1, optimal !

  12. Systematic search for the proof of optimality maximize x1 + x2 x1 + 2x2 3 * y1 4x1 + x2 5 * y2 x1  0 x2  0

  13. Systematic search for the proof of optimality maximize x1 + x2 x1 + 2x2 3 * y1 4x1 + x2 5 * y2 x1  0 x2  0 y1 0 y2 0

  14. Systematic search for the proof of optimality maximize x1 + x2 x1 + 2x2 3 * y1 4x1 + x2 5 * y2 x1  0 x2  0 min 3y1+5y2 y1 0 y2 0 y1 + 4y2 1 2y1+y2 1

  15. Systematic search for the proof of optimality max x1+x2 x1 + 2x2 3 4x1 + x2 5 x1  0 x2  0 min 3y1+5y2 y1 + 4y2 1 2y1+y2 1 y1 0 y2 0 dual linear programs

  16. Systematic search for the proof of optimality max x1+x2 x1 + 2x2 3 4x1 + x2 5 x1  0 x2  0 min 3y1+5y2  y1 + 4y2 1 2y1+y2 1 y1 0 y2 0 dual linear programs

  17. Linear programming duality max x1+x2 x1 + 2x2 3 4x1 + x2 5 x1  0 x2  0 min 3y1+5y2 = y1 + 4y2 1 2y1+y2 1 y1 0 y2 0

  18. Linear programs variables: x1,x2,...,xn linear function: a1x1 + a2x2 + ... + anxn linear constraint: equality a1x1 + a2x2 + ... + anxn = b inequality a1x1 + a2x2 + ... + anxn b

  19. Linear programs variables: x1,x2,...,xn linear function: a1x1 + a2x2 + ... + anxn linear constraint: equality a1x1 + a2x2 + ... + anxn = b inequality a1x1 + a2x2 + ... + anxn b max/min of a linear function subject to collection of linear constraints

  20. Linear programs variables: x1,x2,...,xn linear function: a1x1 + a2x2 + ... + anxn max/min of a linear function subject to collection of linear constraints linear constraint: equality a1x1 + a2x2 + ... + anxn = b inequality a1x1 + a2x2 + ... + anxn b Goal: find the optimal solution (i.e., a feasible solution with the maximum value of the objective)

  21. Linear programs one of the most important modeling tools oil industry manufacturing marketing circuit design very important in theory as well

  22. Shortest path t 4 v 6 2 1 s w 5 3 u

  23. Shortest path t 4 v 6 2 1 s w 5 3 u ds = 0 du ds + 5 dv ds + 6 dw  du + 3 dw  dv + 1 dt  dw + 2 dt  dv + 4 max dt

  24. Max-Flow FLOW CONSERVATION CAPACITY CONSTRAINTS  fu,v = 0 vV fu,v c(u,v) SKEW SYMMETRY fu,v = - fv,u

  25.  fu,v = 0 vV Max-Flow objective = ? us,t: fu,v c(u,v) fu,v + fv,u=0

  26. max  fs,v  fu,v = 0 vV vV Max-Flow us,t: fu,v c(u,v) fu,v + fv,u=0

  27. Linear programming duality maximize  minimize constraint  variable equality  unrestricted  non-negative variable  constraint unrestricted  equality non-negative 

  28. Linear programming duality max x1+x2 x1+x2+x3+x4=1 x1+2x3  1 x2+2x4  2 x1  0 x4  0 maximize  minimize constraint  variable equality  unrestricted  non-negative variable  constraint unrestricted  equality non-negative 

  29. Linear programming duality max x1+x2 x1+x2+x3+x4=1 x1+2x3  1 x2+2x4  2 x1  0 x4  0 y1 y2  0 y3  0 maximize  minimize constraint  variable equality  unrestricted  non-negative variable  constraint unrestricted  equality non-negative  DONE

  30. Linear programming duality max x1+x2 x1+x2+x3+x4=1 x1+2x3  1 x2+2x4  2 x1  0 x4  0 min y1 + y2 + 2 y3 y1 y2  0 y3  0 DONE maximize  minimize constraint  variable equality  unrestricted  non-negative variable  constraint unrestricted  equality non-negative  DONE

  31. Linear programming duality max x1+x2 x1+x2+x3+x4=1 x1+2x3  1 x2+2x4  2 x1  0 x4  0 min y1 + y2 + 2 y3 y1 y2  0 y3  0 DONE maximize  minimize constraint  variable equality  unrestricted  non-negative variable  constraint unrestricted  equality non-negative  DONE y1 + y2 1 y1 + y3 = 1 y1 + 2y2 = 0 y1 + 2y3 0 DONE

  32. Linear programming duality max x1+x2 x1+x2+x3+x4=1 x1+2x3  1 x2+2x4  2 x1  0 x4  0 min y1 + y2 + 2 y3 y2  0 y3  0 y1 + y2 1 y1 + y3 = 1 y1 + 2y2 = 0 y1 + 2y3 0

  33. “” “=” and non-negativity a1 x1 + ... + an xn b a1 x1 + ... + an xn= b + y, y  0 a1 x1 + ... + an xn – y = b, y  0

  34. “=” “” a1 x1 + ... + an xn= b a1 x1 + ... + an xn b a1 x1 + ... + an xnb a1 x1 + ... + an xn b -a1 x1 - ... - an xn -b

  35. optimization  feasibility max a1x1+...+anxn a1x1+...+anxn  P + binary search on P

  36. max  fs,v  fu,v = 0 vV vV Max-Flow us,t: fu,v c(u,v) fu,v + fv,u=0

  37. max  fs,v  fu,v = 0 vV vV Max-Flow us,t: yu zu,v  0 fu,v c(u,v) fu,v + fv,u=0 w{u,v}

  38. max  fs,v  fu,v = 0 vV vV Max-Flow min  c(u,v)zu,v u,v us,t: yu fu,v c(u,v) zu,v fu,v + fv,u=0 w{u,v} zu,v  0

  39. max  fs,v  fu,v = 0 vV vV Max-Flow min  c(u,v)zu,v u,v us,t us,t: yu + fu,v c(u,v) zu,v =0 + fu,v + fv,u=0 w{u,v} zu,v  0

  40. Max-Flow min  c(u,v)zu,v u,v yu + zu,v + w{u,v} =0 us,t ys = -1 zs,v + w{s,v} =1 yt = 0 zt,v + w{t,v} =0 zu,v  0

  41. Max-Flow min  c(u,v)zu,v u,v yu + zu,v + w{u,v} =0 ys = -1 yt = 0 zu,v  0

  42. Max-Flow min  c(u,v)zu,v u,v yu + zu,v + w{u,v} =0 yv + zv,u + w{u,v} =0 ys = -1 yt = 0 zu,v  0

  43. Max-Flow min  c(u,v)zu,v u,v yu + zu,v + w{u,v} =0 yv + zv,u + w{u,v} =0 ys = -1 yt = 0 yu - yv= zv,u - zu,v zu,v  0

  44. min  c(u,v)zu,v u,v Max-Flow ys = -1 yu - yv= zv,u - zu,v yt = 0 zu,v  0

  45. min  c(u,v) max{0,yu-yv} u,v Max-Flow ys = -1 yu - yv= zv,u - zu,v yt = 0 zu,v  0

  46. min  c(u,v) max{0,yu-yv} u,v Max-Flow ys = -1 yt = 0

  47. min  c(u,v) max{0,yu-yv} u,v Max-Flow = Min-Cut ys = -1 one more trick achieves yu {-1,0} yt = 0 min  c(u,v) S,s S tSC u S,v SC

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