Lecture3

Lecture3 - C&O 355 Lecture 3 N. Harvey

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: C&O 355 Lecture 3 N. Harvey http://www.math.uwaterloo.ca/~harvey/ Outline • Review Local-Search Algorithm • Pitfall #1: Defining corner points – Polyhedra that don’t contain a line have corner points • Pitfall #2: No corner points? – Equational form of LPs Local-Search Algorithm: Pitfalls & Details 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 Local-Search Algorithm: Pitfalls & Details 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 1. What is a corner point? 2. What if there are no corner points? 3. What are the “neighboring” corner points? 4. How to choose a neighboring point? 5. How can I find a starting corner point? 6. Does the algorithm terminate? 7. Does it produce the right answer? Pitfall #1: What is a corner point? • How should we define corner points? • Under any reasonable definition, point x should be considered a corner point x Pitfall #1: What is a corner point? • Attempt #1: “x is the ‘farthest point’ in some direction” • Let P = { feasible region } • There exists c 2 R n s.t. c T x>c T y for all y 2 P n {x} • “For some objective function, x is the unique optimal point when maximizing over P” • Such a point x is called a “ vertex ” c x is unique optimal point Pitfall #1: What is a corner point? • Attempt #2: “There is no feasible line-segment that goes through x in both directions” • Whenever x= ® y+(1- ® )z with y,z x and ® 2 (0,1), then either y or z must be infeasible. • “If you write x as a convex combination of two feasible points y and z, the only possibility is x=y=z” • Such a point x is called an “ extreme point ” y z (infeasible) x Pitfall #1: What is a corner point? • Attempt #3: “x lies on the boundary of many constraints” • Note: This discussion differs from textbook x lies on boundary of two constraints x 4x 1- x 2 · 10 x 1 + 6x 2 · 15 Pitfall #1: What is a corner point? • Attempt #3: “x lies on the boundary of many constraints” • Note: This discussion differs from textbook • What if I introduce redundant constraints? y also lies on boundary of two constraints y Not the right condition x 1 + 6x 2 · 15 2x 1 + 12x 2 · 30 Pitfall #1: What is a corner point? • Revised Attempt #3: “x lies on the boundary of many linearly independent constraints” • Feasible region: P = { x : a i T x · b i 8 i } ½ R n • Let I x ={ i : a i T x=b i } and A x ={ a i : i 2I x }. (“ Tight constraints ”) • x is a “ basic feasible solution (BFS) ” if rank A x = n y x 1 + 6x 2 · 15 2x 1 + 12x 2 · 30 x y’s constraints are linearly dependent 4x 1- x 2 · 10 x’s constraints are linearly independent x 1 + 6x 2 · 15 Lemma : Let P be a polyhedron. The following are equivalent....
View Full 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 / 29

Lecture3 - C&O 355 Lecture 3 N. Harvey

This preview shows document pages 1 - 12. Sign up to view the full document.

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