{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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
C&O 355 Lecture 3 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 Document Right Arrow Icon
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
Background image of page 2
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
Background image of page 3

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

View Full Document Right Arrow Icon
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?
Background image of page 4
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
Background image of page 5

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

View Full Document Right Arrow Icon
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
Background image of page 6
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
Background image of page 7

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

View Full Document Right Arrow Icon
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
Background image of page 8
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
Background image of page 9

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

View Full Document Right Arrow Icon
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
Background image of page 10
Lemma : Let P be a polyhedron. The following are equivalent.
Background image of page 11

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

View Full Document Right Arrow Icon
Image of page 12
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}