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Lecture3

# Lecture3 - C&O 355 Lecture 3 N Harvey...

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C&O 355 Lecture 3 N. Harvey http://www.math.uwaterloo.ca/~harvey/

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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

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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

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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

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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

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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.

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