Lecture4

Lecture4 - C&O 355 Lecture 4 N Harvey http/www.math.uwaterloo.ca/~harvey Outline Equational form of LPs Basic Feasible Solutions for Equational

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
C&O 355 Lecture 4 N. Harvey http://www.math.uwaterloo.ca/~harvey/
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Outline Equational form of LPs Basic Feasible Solutions for Equational form LPs Bases and Feasible Bases Brute-Force Algorithm Neighboring Bases
Background image of page 2
Local-Search Algorithm: Pitfalls & Details 1. What is a corner point? 2. What if there are no corner points? 3. What are the “neighboring” corner points? 4. What if there are no neighboring corner points? 5. How can I find a starting corner point? 6. Does the algorithm terminate? 7. Does it produce the right answer? Algorithm Let x be any corner point For each corner point y that is a neighbor of x If c T y>c T x then set x=y Halt
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Pitfall #2: No corner points? This is possible Case 1: LP infeasible Case 2: Not enough constraints x 1 x 2 x 2 · 2 x 2 ¸ 0 This is unavoidable. Algorithm must detect this case. A Fix! We avoid this case by using equational form. x 1 x 2 x 2 - x 1 ¸ 1 x 1 + 6x 2 · 15 4x 1 - x 2 ¸ 10 (0,0) x 1 ¸ 0 x 2 ¸ 0
Background image of page 4
Converting to Equational Form “Inequality form” Equational form” Claim : These two forms of LPs are equivalent. x 1 x 2 x 2 - x 1 · 1 x 1 + 6x 2 · 15 4x 1 - x 2 · 10 (0,0) x 1 ¸ 0 x 2 ¸ 0 (3,2) x 1 x 2 Solutions of Ax=b Feasible region x 3 A · x b A = x b Tall, skinny A Short, wide A
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Easy: Just use “Simple LP Manipulations” from Lecture 2 Trick 1: ¸ ” instead of “ · Trick 2: “=” instead of “ · This shows P={ x : Ax=b, x ¸ 0 } is a polyhedron . “Inequality form” Equational form”
Background image of page 6
Trick 1: x 2 R can be written x=y-z where y,z ¸ 0 So Trick 2: For u,v 2 R , u · v , 9 w ¸ 0 s.t. u+w=v So Rewrite it: Then “Inequality form” Equational form” ´ ´ ~ A = [ A; ¡ A;I ] ~ c = [ c; ¡ c; 0] ~ x = [ y;z;w ] ´ “slack variable” This is already in equational form!
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Pitfall #2: No corner points?
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/16/2011 for the course C 355 taught by Professor Harvey during the Fall '09 term at Waterloo.

Page1 / 19

Lecture4 - C&O 355 Lecture 4 N Harvey http/www.math.uwaterloo.ca/~harvey Outline Equational form of LPs Basic Feasible Solutions for Equational

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online