agenda
Download
Skip this Video
Download Presentation
Agenda

Loading in 2 Seconds...

play fullscreen
1 / 11

Agenda - PowerPoint PPT Presentation


  • 134 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Agenda' - mili


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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)

ad