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# Agenda - PowerPoint PPT Presentation

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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Presentation Transcript
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

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:

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
• 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

maxx pTx

s.t. Ax <= c

x >= 0

equivalent to

miny cTy

s.t. ATy >= p

y >= 0

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 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 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
• min f(x) = - max -f(x)
• g(x) <= b same as -g(x) >= -b
• x <= 5 same as -x >= -5
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

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)