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

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C&O 355 Lecture 7 N. Harvey http://www.math.uwaterloo.ca/~harvey/
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Outline Finding a starting point Two small issues Duality Geometric view Algebraic view Dual LP & Weak Duality Primal vs Dual Strong Duality Theorem Certificates
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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
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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 ¸ 0 (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 , 0 ) 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 )=( 0 , b ) is a trivial solution!
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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 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
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Two Small Issues
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