Agenda Duality Geometric Picture Piecewise linear functions

Dual Problem Original: maxprofit from running plant s.t.capacity not exceeded variables are production quantities Dual: mincost to buy all capacity s.t.willing to sell capacity instead of produce variables are prices

Dual Problem Original: max$840 profit * S cars + … s.t.3hr * S + 2hr * F + 1hr * L <= 120hr engine shop capacity 1hr * S + 2hr * F + 3hr * L <= 80hr body shop capacity … variables S, F, L are production quantities Dual: minprice E * 120 hr engine shop capacity + … s.t.3hr * E + 1hr * B + 2hr * SF >= $840 (standard car profit) 2hr * E + 2hr * B + 3hr * FF >= $1120 (fancy car profit) … variables E, B, SF, FF, FL are prices

Results constraint becomes dual variable –constraint bound goes into dual objective –shadow price = optimal dual variable variable becomes dual constraint –objective coefficient is dual constraint bound –optimal value = dual shadow price max problem becomes min problem solutions the same –unbounded problem becomes infeasible

Generic Dual Problem max x p T x s.t. Ax <= c x >= 0 equivalent to min y c T y s.t.A T y >= p y >= 0

Electric Utility Example Customer demand d Generator i has cost c i and capacity b i Production x i on generator i Goal: meet demand with little cost min x c T x s.t.x 1 +x 2 +…+x n >= d x i <= b i for i=1,..,n x >= 0

Electric Utility Example Original: min x c T x s.t.x 1 +x 2 +…+x n >= d x i <= b i for i=1,…,n x >= 0 Dual: max p,y dp - b T y s.t.p - y i <= c i for i=1,…,n p >= 0, y >= 0

Electric Utility Example Dual max p,y dp - b T y s.t.p - y i <= c i for i=1,…,n p >= 0, y >= 0 p = market price for power y i = profit rate at generator i constraint: y i >= p - c i Goal: max net revenue (after paying out-sourced generators their profit)

Manipulations min f(x) = - max -f(x) g(x) = -b x = -5

General Dual Formulation for max problem = 0 >= constraint becomes variable <= 0 = constraint becomes variable without bound for min problem the opposite max x p T x s.t.Ax ? c x ? 0 min y c T y s.t.A T y ? p y ? 0

Piecewise Linear Functions min x c 1 (x 1 ) + c 2 x 2 s.t.x 1 +x 2 >= d x >= 0 min x,z z + c 2 x 2 s.t.x 1 +x 2 >= d x >= 0 z >= s 1 x 1 z >= s 2 x 1 + t c 1 (x 1 )