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Unformatted text preview: C&O 355 Lecture 3 N. Harvey http://www.math.uwaterloo.ca/~harvey/ Outline • Review LocalSearch 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 LocalSearch 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 LocalSearch 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 linesegment 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....
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This note was uploaded on 06/16/2011 for the course C 355 taught by Professor Harvey during the Fall '09 term at Waterloo.
 Fall '09
 Harvey

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