1 / 11

# Agenda

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

Télécharger la présentation

## Agenda

An Image/Link below is provided (as is) to download presentation Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

### Presentation Transcript

1. Agenda • Duality • Geometric Picture • Piecewise linear functions

2. 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

3. 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

4. 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

5. Generic Dual Problem maxx pTx s.t. Ax <= c x >= 0 equivalent to miny cTy s.t. ATy >= p y >= 0

6. 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

7. 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

8. 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)

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

10. 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

11. 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)

More Related