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 x1x1 x2x2

Linear programming maximize x 1 + x 2 x 1 + 3x 2  3 3x 1 + x 2  5 x 1  0 x 2  0 x1x1 x2x2

Linear programming maximize x 1 + x 2 x 1 + 3x 2  3 3x 1 + x 2  5 x 1  0 x 2  0 x1x1 x2x2 feasible solutions

Linear programming maximize x 1 + x 2 x 1 + 3x 2  3 3x 1 + x 2  5 x 1  0 x 2  0 x1x1 x2x2 optimal solution x 1 =1/2, x 2 =3/2

Can you prove it is optimal ? maximize x 1 + x 2 x 1 + 3x 2  3 3x 1 + x 2  5 x 1  0 x 2  0 x1x1 x2x2 optimal solution x 1 =1/2, x 2 =3/2

Can you prove it is optimal ? maximize x 1 + x 2 x 1 + 3x 2  3 3x 1 + x 2  5 4x 1 + 4x 2  8 x1x1 x2x2 optimal solution x 1 =1/2, x 2 =3/2

Can you prove it is optimal ? maximize x 1 + x 2 x 1 + 3x 2  3 3x 1 + x 2  5 x 1 +x 2  2 x1x1 x2x2 optimal solution x 1 =1/2, x 2 =3/2

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

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

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

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

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

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

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

Linear programming duality max x 1 +x 2 x 1 + 2x 2  3 4x 1 + x 2  5 x 1  0 x 2  0 y 1  0 y 2  0 min 3y 1 +5y 2 y 1 + 4y 2  1 2y 1 +y 2  1 

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

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

Linear programs variables: x 1,x 2,...,x n linear function: a 1 x 1 + a 2 x a n x n linear constraint: equality a 1 x 1 + a 2 x a n x n = b inequality a 1 x 1 + a 2 x a n x n  b max/min of a linear function subject to collection of linear constraints 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 s t u v w

s t u v w d s = 0 d u  d s + 5 d v  d s + 6 d w  d u + 3 d w  d v + 1 d t  d w + 2 d t  d v + 4 max d t

Max-Flow FLOW CONSERVATION CAPACITY CONSTRAINTS  f u,v = 0 vVvV f u,v  c(u,v) SKEW SYMMETRY f u,v = - f v,u

Max-Flow  f u,v = 0 vVvV f u,v  c(u,v) f u,v + f v,u =0 objective = ? u  s,t:

Max-Flow  f u,v = 0 vVvV f u,v  c(u,v) f u,v + f v,u =0 max  f s,v vVvV u  s,t:

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

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

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

Linear programming duality maximize  minimize constraint  variable equality  unrestricted   non-negative variable  constraint unrestricted  equality non-negative   max x 1 +x 2 x 1 +x 2 +x 3 +x 4 =1 x 1 +2x 3  1 x 2 +2x 4  2 x 1  0 x 4  0 y 1 y 2  0 y 3  0 min y 1 + y y 3 DONE

Linear programming duality maximize  minimize constraint  variable equality  unrestricted   non-negative variable  constraint unrestricted  equality non-negative   max x 1 +x 2 x 1 +x 2 +x 3 +x 4 =1 x 1 +2x 3  1 x 2 +2x 4  2 x 1  0 x 4  0 y 1 y 2  0 y 3  0 min y 1 + y y 3 DONE y 1 + y 2  1 y 1 + y 3 = 1 y 1 + 2y 2 = 0 y 1 + 2y 3  0 DONE

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

a 1 x a n x n  b a 1 x a n x n  b + y, y  0 a 1 x a n x n – y  b, y  0  “  ”   “=” and non-negativity

a 1 x a n x n  b a 1 x a n x n  b a 1 x a n x n  b  “  ”   “  ” a 1 x a n x n  b -a 1 x a n x n  -b

optimization  feasibility max a 1 x a n x n a 1 x a n x n  P + binary search on P

Max-Flow  f u,v = 0 vVvV f u,v  c(u,v) f u,v + f v,u =0 max  f s,v vVvV u  s,t:

Max-Flow  f u,v = 0 vVvV f u,v  c(u,v) f u,v + f v,u =0 max  f s,v vVvV yuyu z u,v  0 w {u,v} u  s,t:

Max-Flow  f u,v = 0 vVvV f u,v  c(u,v) f u,v + f v,u =0 max  f s,v vVvV yuyu z u,v w {u,v} min  c(u,v)z u,v u,v u  s,t: z u,v  0

Max-Flow  f u,v = 0 vVvV f u,v  c(u,v) f u,v + f v,u =0 max  f s,v vVvV yuyu z u,v w {u,v} min  c(u,v)z u,v u,v + + =0 u  s,t u  s,t: z u,v  0

Max-Flow min  c(u,v)z u,v u,v u  s,t y u + z u,v + w { u,v } =0 z s,v + w { s,v } =1 z t,v + w { t,v } =0 z u,v  0 y s = -1 y t = 0

Max-Flow min  c(u,v)z u,v u,v y u + z u,v + w { u,v } =0 z u,v  0 y s = -1 y t = 0

Max-Flow min  c(u,v)z u,v u,v y u + z u,v + w { u,v } =0 z u,v  0 y s = -1 y t = 0 y v + z v,u + w { u,v } =0

Max-Flow min  c(u,v)z u,v u,v y u + z u,v + w { u,v } =0 z u,v  0 y s = -1 y t = 0 y v + z v,u + w { u,v } =0 y u - y v = z v,u - z u,v

Max-Flow min  c(u,v)z u,v u,v z u,v  0 y s = -1 y t = 0 y u - y v = z v,u - z u,v

Max-Flow min  c(u,v) max{0,y u -y v } u,v z u,v  0 y s = -1 y t = 0 y u - y v = z v,u - z u,v

Max-Flow min  c(u,v) max{0,y u -y v } u,v y s = -1 y t = 0

Max-Flow = Min-Cut min  c(u,v) max{0,y u -y v } u,v y s = -1 y t = 0 min  c(u,v) u  S,v  S C S,s  S t  S C one more trick achieves y u  {-1,0}