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Agenda. Duality Geometric Picture Piecewise linear functions. Dual Problem. Original: max profit from running plant s.t. capacity not exceeded variables are production quantities Dual: min cost to buy all capacity s.t. willing to sell capacity instead of produce

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Agenda
Agenda

  • Duality

  • Geometric Picture

  • Piecewise linear functions


Dual problem
Dual Problem

Original:

max profit from running plant

s.t. capacity not exceeded

variables are production quantities

Dual:

min cost to buy all capacity

s.t. willing to sell capacity instead of produce

variables are prices


Dual problem1
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:

min price 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
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
Generic Dual Problem

maxx pTx

s.t. Ax <= c

x >= 0

equivalent to

miny cTy

s.t. ATy >= p

y >= 0


Electric utility example
Electric Utility Example

  • Customer demand d

  • Generator i has cost ci and capacity bi

  • Production xi on generator i

  • Goal: meet demand with little cost

    minx cTx

    s.t. x1+x2+…+xn >= d

    xi <= bi for i=1,..,n

    x >= 0


Electric utility example1
Electric Utility Example

Original:

minx cTx

s.t. x1+x2+…+xn >= d

xi <= bi for i=1,…,n

x >= 0

Dual:

maxp,y dp - bTy

s.t. p - yi <= ci for i=1,…,n

p >= 0, y >= 0


Electric utility example2
Electric Utility Example

Dual

maxp,y dp - bTy

s.t. p - yi <= ci for i=1,…,n

p >= 0, y >= 0

p = market price for power

yi = profit rate at generator i

constraint: yi >= p - ci

Goal: max net revenue

(after paying out-sourced generators their profit)


Manipulations
Manipulations

  • min f(x) = - max -f(x)

  • g(x) <= b same as -g(x) >= -b

  • x <= 5 same as -x >= -5


General dual formulation
General Dual Formulation

maxx pTx

s.t. Ax ? c

x ? 0

miny cTy

s.t. ATy ? p

y ? 0

  • for max problem

    <= constraint becomes variable >= 0

    >= constraint becomes variable <= 0

    = constraint becomes variable without bound

  • for min problem the opposite


Piecewise linear functions
Piecewise Linear Functions

minx c1(x1) + c2x2

s.t. x1+x2 >= d

x >= 0

minx,z z + c2x2

s.t. x1+x2 >= d

x >= 0

z >= s1 x1

z >= s2 x1 + t

c1(x1)


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