Linear programming maximize x 1 x 2 x

Linear programming maximize x 1 + x 2 x 1 + 3 x 2 3 3 x 1 + x 2 5 x 1 0 x 2 0

Linear programming maximize x 1 + x 2 x 1 + 3 x 2 3 3 x 1 + x 2 5 x 1 0 x 2

Linear programming maximize x 1 + x 2 x 1 + 3 x 2 3 3 x 1 + x 2 5 x 1 0 x 2

Linear programming maximize x 1 + x 2 x 1 feasible solutions x 1 + 3 x 2 3 3 x 1 + x 2 5 x 1 0 x 2

Linear programming maximize x 1 + x 2 x 1 optimal solution x 1=1/2, x 2=3/2 x 1 + 3 x 2 3 3 x 1 + x 2 5 x 1 0 x 2

Can you prove it is optimal ? maximize x 1 + x 2 x 1 optimal solution x 1=1/2, x 2=3/2 x 1 + 3 x 2 3 3 x 1 + x 2 5 x 1 0 x 2

Can you prove it is optimal ? maximize x 1 + x 2 x 1 optimal solution x 1=1/2, x 2=3/2 x 1 + 3 x 2 3 3 x 1 + x 2 5 4 x 1 + 4 x 2 8 x 2

Can you prove it is optimal ? maximize x 1 + x 2 x 1 optimal solution x 1=1/2, x 2=3/2 x 1 + 3 x 2 3 3 x 1 + x 2 5 x 1+x 2 2 x 2

Another linear program maximize x 1 + x 2 x 1 + 2 x 2 3 4 x 1 + x 2 5 x 1 0 x 2 0

Another linear program maximize x 1 + x 2 x 1 + 2 x 2 3 4 x 1 + x 2 5 x 1 0 x 2 0 x 1=1, x 2=1, optimal ?

Another linear program maximize x 1 + x 2 x 1 + 2 x 2 3 4 x 1 + x 2 5 x 1 0 x 2 0 x 1=1, x 2=1, *3 *1 7 x 1 + 7 x 2 14 optimal !

Systematic search for the proof of optimality maximize x 1 + x 2 x 1 + 2 x 2 3 4 x 1 + x 2 5 x 1 0 x 2 0 * y 1 * y 2

Systematic search for the proof of optimality maximize x 1 + x 2 x 1 + 2 x 2 3 4 x 1 + x 2 5 x 1 0 x 2 0 * y 1 * y 2 y 1 0 y 2 0

Systematic search for the proof of optimality maximize x 1 + x 2 x 1 + 2 x 2 3 * y 1 4 x 1 + x 2 5 * y 2 x 1 0 x 2 0 min 3 y 1+5 y 2 y 1 + 4 y 2 1 2 y 1+y 2 1 y 1 0 y 2 0

Systematic search for the proof of optimality max x 1+x 2 min 3 y 1+5 y 2 x 1 + 2 x 2 3 4 x 1 + x 2 5 x 1 0 x 2 0 y 1 + 4 y 2 1 2 y 1+y 2 1 y 1 0 y 2 0 dual linear programs

Systematic search for the proof of optimality max x 1+x 2 x 1 + 2 x 2 3 4 x 1 + x 2 5 x 1 0 x 2 0 min 3 y 1+5 y 2 y 1 + 4 y 2 1 2 y 1+y 2 1 y 1 0 y 2 0 dual linear programs

Linear programming duality max x 1+x 2 x 1 + 2 x 2 3 4 x 1 + x 2 5 x 1 0 x 2 0 = min 3 y 1+5 y 2 y 1 + 4 y 2 1 2 y 1+y 2 1 y 1 0 y 2 0

Linear programs variables: x 1, x 2, . . . , xn linear function: a 1 x 1 + a 2 x 2 +. . . + anxn linear constraint: equality a 1 x 1 + a 2 x 2 +. . . + anxn = b inequality a 1 x 1 + a 2 x 2 +. . . + anxn b

Linear programs variables: x 1, x 2, . . . , xn linear function: a 1 x 1 + a 2 x 2 +. . . + anxn linear constraint: equality a 1 x 1 + a 2 x 2 +. . . + anxn = b inequality a 1 x 1 + a 2 x 2 +. . . + anxn b max/min of a linear function subject to collection of linear constraints

Linear programs max/min of a linear function subject to collection of linear constraints variables: x 1, x 2, . . . , xn linear function: a 1 x 1 + a 2 x 2 +. . . + anxn linear constraint: equality a 1 x 1 + a 2 x 2 +. . . + anxn = b inequality a 1 x 1 + a 2 x 2 +. . . + anxn b Goal: find the optimal solution (i. e. , a feasible solution with the maximum value of the objective)

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

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

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

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

Max-Flow objective = ? u s, t: fu, v = 0 v V fu, v c(u, v) fu, v + fv, u=0

Max-Flow max fs, v v V u s, t: fu, v = 0 v V fu, v c(u, v) fu, v + fv, u=0

Linear programming duality maximize minimize constraint variable equality unrestricted non-negative variable constraint unrestricted equality non-negative

Linear programming duality max x 1+x 2+x 3+x 4=1 x 1+2 x 3 1 x 2+2 x 4 2 x 1 0 x 4 0 maximize minimize constraint variable equality unrestricted non-negative variable constraint unrestricted equality non-negative

Linear programming duality y 1 y 2 0 y 3 0 NE maximize minimize DO max x 1+x 2+x 3+x 4=1 x 1+2 x 3 1 x 2+2 x 4 2 x 1 0 x 4 0 constraint variable equality unrestricted non-negative variable constraint unrestricted equality non-negative

Linear programming duality NE maximize minimize DO y 1 y 2 0 y 3 0 NE min y 1 + y 2 + 2 y 3 DO max x 1+x 2+x 3+x 4=1 x 1+2 x 3 1 x 2+2 x 4 2 x 1 0 x 4 0 constraint variable equality unrestricted non-negative variable constraint unrestricted equality non-negative

Linear programming duality NE NE NE maximize minimize DO y 1 y 2 0 y 3 0 DO y 1 + y 2 1 y 1 + y 3 = 1 y 1 + 2 y 2 = 0 y 1 + 2 y 3 0 min y 1 + y 2 + 2 y 3 DO max x 1+x 2+x 3+x 4=1 x 1+2 x 3 1 x 2+2 x 4 2 x 1 0 x 4 0 constraint variable equality unrestricted non-negative variable constraint unrestricted equality non-negative

Linear programming duality max x 1+x 2+x 3+x 4=1 x 1+2 x 3 1 x 2+2 x 4 2 x 1 0 x 4 0 min y 1 + y 2 + 2 y 3 y 2 0 y 3 0 y 1 + y 2 1 y 1 + y 3 = 1 y 1 + 2 y 2 = 0 y 1 + 2 y 3 0

“ ” “=” and non-negativity a 1 x 1 +. . . + an xn b a 1 x 1 +. . . + an xn = b + y, y 0 a 1 x 1 +. . . + an xn – y = b, y 0

optimization feasibility max a 1 x 1+. . . +anxn P + binary search on P

Max-Flow max fs, v v V u s, t: fu, v = 0 v V fu, v c(u, v) fu, v + fv, u=0

Max-Flow max fs, v v V u s, t: fu, v = 0 v V yu fu, v c(u, v) zu, v 0 fu, v + fv, u=0 w{u, v}

Max-Flow min c(u, v)zu, v max fs, v v V u s, t: fu, v = 0 v V yu fu, v c(u, v) zu, v fu, v + fv, u=0 w{u, v} zu, v 0

Max-Flow min c(u, v)zu, v max fs, v v V u s, t: fu, v = 0 v V yu fu, v c(u, v) zu, v =0 + w{u, v} fu, v + fv, u=0 z u s, t 0 +

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

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

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

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

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

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

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

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