Lecture7

Lecture7 - C&O 355 Lecture 7 N Harvey...

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Unformatted text preview: C&O 355 Lecture 7 N. Harvey http://www.math.uwaterloo.ca/~harvey/ Outline • Finding a starting point • Two small issues • Duality – Geometric view – Algebraic view – Dual LP & Weak Duality – Primal vs Dual – Strong Duality Theorem – Certificates 1. What is a corner point? (BFS and bases) 2. What if there are no corner points? (Infeasible) 3. What are the “neighboring” bases? (Increase one coordinate) 4. What if no neighbors are strictly better? (Might move to a basis that isn’t strictly better (if ± =0), but whenever x changes it’s strictly better) 5. How can I find a starting feasible basis? 6. Does the algorithm terminate? (If Bland’s rule used) 7. Does it produce the right answer? (Yes) Local-Search Algorithm Let B be a feasible basis (If none, Halt: LP is infeasible) For each entering coordinate k B If “benefit” of coordinate k is > 0 Compute y( ± ) (If ± = 1 , Halt: LP is unbounded) Find leaving coordinate h 2 B (y( ± ) h =0) Set x=y( ± ) and B’= B n {h} [ {k} Halt: return x Finding a starting point • Consider LP max { c T x : x 2 P } where P={ x : A x =b, x ¸ 0 } • How can we find a feasible point? • Trick: Just solve a different LP! – Note: c is irrelevant. We can introduce a new objective function – WLOG, b ¸ (Can multiply constraints by -1) – Allow “A x =b” constraint to be violated via “artificial variables” : Q = { ( x , y ) : A x + y =b, x ¸ 0, y ¸ 0 } – Note: ( x , ) 2 Q , x 2 P. Can we find such a point? – Solve the new LP min { § i y i : ( x , y ) 2 Q } – If the optimal value is 0, then x 2 P. If not, P is empty! – How do we find feasible point for the new LP? • ( x , y )=( , b ) is a trivial solution! 1. What is a corner point? (BFS and bases) 2. What if there are no corner points? (Infeasible) 3. What are the “neighboring” bases? (Increase one coordinate) 4. What if no neighbors are strictly better? (Might move to a basis that isn’t strictly better (if ± =0), but whenever x changes it’s strictly better) 5. How can I find a starting feasible basis? (Solve an easier LP) 6. Does the algorithm terminate? (If Bland’s rule used) 7. Does it produce the right answer? (Yes) Local-Search Algorithm Let B be a feasible basis (If none, Halt: LP is infeasible) For each entering coordinate k B If “benefit” of coordinate k is > 0...
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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.

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Lecture7 - C&O 355 Lecture 7 N Harvey...

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