ISE 536Fall03: Linear Programming and Extensions
October 13, 2003
Lecture 12: Geometry of the Dual
Lecturer: Fernando Ordnez o~
1
Review Duality
For any linear programming problem (P ) min ct x s.t. Ax b x0 that satisfies: Weak duality: For any x feasible
ISE 536Fall03: Linear Programming and Extensions
August 27, 2003
Lecture 1: Introduction to Linear Programming
Lecturer: Fernando Ordnez o~
1
Course Organization
Instructor: Fernando Ordnez o~ GER-247, x1-2413 [email protected] Office hours: 1:00-3:00 p.m. o
ISE 536Fall03: Linear Programming and Extensions
September 3, 2003
Lecture 2: Intro. Examples and Transformations
Lecturer: Fernando Ordnez o~
1
1.1
More LP Examples
Investment
Suppose that 5 investment alternatives are described by the following table, w
ISE 536Fall03: Linear Programming and Extensions
September 3, 2003
Lecture 3: Geometry, Basic Feasible Solutions
Lecturer: Fernando Ordnez o~
1
Examples of LP Geometry
Let us construct examples of LPs which have bounded feasible region and single solution
ISE 536Fall03: Linear Programming and Extensions
September 3, 2003
Lecture 4: Geometry, BFS in Standard Form
Lecturer: Fernando Ordnez o~
1
BFS vertex extreme point
Let us recall the definitions x is an extreme point of a polyhedron P if it can't be expre
ISE 536Fall03: Linear Programming and Extensions
September 15, 2003
Lecture 5: Geometry, Optimality of BFS
Lecturer: Fernando Ordnez o~
1
Existence of a BFS
n
Theorem 1. P = cfw_x Proof: pg. 63-64.
| Ax b = . P has a BFS iff P does not contain a line
Chec
ISE 536Fall03: Linear Programming and Extensions
September 17, 2003
Lecture 6: Simplex Method, Moving in the Polyhedra
Lecturer: Fernando Ordnez o~
1
Conceptual Algorithm to solve LP
n
Consider the following to solve min ct x | x P , where P = cfw_x 1. 2.
ISE 536Fall03: Linear Programming and Extensions
September 22, 2003
Lecture 7: Simplex Method, The Tableau
Lecturer: Fernando Ordnez o~
1
Outline of Algorithm
The makings of the Simplex algorithm, and formulas to keep in mind: Find a BFS Ax = b x0 A = [BN
ISE 536Fall03: Linear Programming and Extensions
September 24, 2003
Lecture 8: Simplex Method, Optimality Conditions
Lecturer: Fernando Ordnez o~
1
1.1
The Tableau
Reminder
A BFS: xB = B -1 b 0, xN = 0. Find a basic direction d: (dN )j = 1, (dN )i = 0, i
ISE 536Fall03: Linear Programming and Extensions
September 24, 2003
Lecture 9: Simplex Method, Degeneracy
Lecturer: Fernando Ordnez o~
1
Full simplex algorithm
To solve min ct x : Ax = b, x 0. To start the algorithm we need a basic solution B -1 b and we
ISE 536Fall03: Linear Programming and Extensions
December 3, 2003
Lecture 25: LP in Research
Lecturer: Fernando Ordnez o~
Robust Capacity Expansion of Transit Networks 1 Capacity expansion problem
We consider the problem of deciding investment decisions o
ISE 536Fall03: Linear Programming and Extensions
December 1, 2003
Lecture 24: Integer Programming, Branch and Bound
Lecturer: Fernando Ordnez o~
1
Branch and Bound
We now present a method used to solve mixed integer programs, such as: z = min ct x + dt y
ISE 536Fall03: Linear Programming and Extensions
November 26, 2003
Lecture 23: Integer Programming, Modeling
Lecturer: Fernando Ordnez o~
1
Mixed Integer Programming Problem
Here we study linear programming problems with some variables constrained to be i
ISE 536Fall03: Linear Programming and Extensions
October 15, 2003
Lecture 13: Dual Simplex, Farkas Lemma
Lecturer: Fernando Ordnez o~
1
Dual Simplex
Usual Simplex Maintain a BFS and aim for optimality (i.e. dual feasibility 0 ct - ct B -1 A = B ct - y t
ISE 536Fall03: Linear Programming and Extensions
October 22, 2003
Lecture 14: Sensitivity Analysis
Lecturer: Fernando Ordnez o~
1
Re-optimizing a problem
In this section we will consider that we solve a problem (P ) min ct x s.t. Ax = b x0 to optimality.
ISE 536Fall03: Linear Programming and Extensions
October 27, 2003
Lecture 15: Sensitivity Analysis, Continued
Lecturer: Fernando Ordnez o~
1
Global Dependence on b
Let P (b) = cfw_x | Ax = b, x 0 , define the set S = cfw_b | P (b) = , and for every b S de
ISE 536Fall03: Linear Programming and Extensions
November 3, 2003
Lecture 17: Review Duality and Sensitivity
Lecturer: Fernando Ordnez o~
Problem 5.8 max 51E+ 102C+ s.t. 10E+ E+ 3E+ 2E+ 2C+ C+ 4C+ 66P1 2P1 + 6P1 + 2P1 + P1 - Solution: Optimal value E C P1
ISE 536Fall03: Linear Programming and Extensions
November 5, 2003
Lecture 18: Large scale LP: Column Generation
Lecturer: Fernando Ordnez o~
1
Example
Consider the problem of operating a supermarket chain. There are a large number of variables that apply
ISE 536Fall03: Linear Programming and Extensions
November 12, 2003
Lecture 19: Bender's Decomposition
Lecturer: Fernando Ordnez o~
1
Motivation: Stochastic Programming
An electric utility company faces the problem of satisfying demand at minimum cost. In
ISE 536Fall03: Linear Programming and Extensions
November 17, 2003
Lecture 20: Large Scale LP, Examples
Lecturer: Fernando Ordnez o~
1
Bounds on Benders Decomposition
Let (x , ) be the optimal solution, and z (x , ) be the optimal objective function value
ISE 536Fall03: Linear Programming and Extensions
November 19, 2003
Lecture 21: Interior Point Methods
Lecturer: Fernando Ordnez o~
1
Complexity of LP
The Klee-Minty example says:
To measure the efficiency of an algorithm, we need: Size of an instance. For
ISE 536Fall03: Linear Programming and Extensions
November 24, 2003
Lecture 22: IPM, Path Following Methods
Lecturer: Fernando Ordnez o~
1
A few ideas from convex optimization
n
For a convex function f :
, the point that minimizes f (x), satisfies
f (x) =
ISE 536Fall03: Linear Programming and Extensions
October 6, 2003
Lecture 11: Duality, Introduction
Lecturer: Fernando Ordnez o~
1
Definition
Here the rows of the matrix A are at , for i = 1, . . . , m, and the columns are Aj , j = 1, . . . , n. i For any