Lecture5

Lecture5 - C&O 355 Lecture 5 N Harvey...

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Unformatted text preview: C&O 355 Lecture 5 N. Harvey http://www.math.uwaterloo.ca/~harvey/ Outline • Neighboring Bases • Finding Better Neighbors • “Benefit” of a coordinate • Quick optimality proof • Alternative optimality proof – Generalized neighbors and generalized benefit 1. What is a corner point? (BFS and bases) 2. What if there are no corner points? (Infeasible) 3. What are the “neighboring” bases? 4. What if no neighbors are strictly better? 5. How can I find a starting feasible basis? 6. Does the algorithm terminate? 7. Does it produce the right answer? Local-Search Algorithm Let B be a feasible basis (if none, infeasible) For each feasible basis B’ that is a neighbor of B Compute BFS y defined by B’ If c T y>c T x then set x=y Halt Neighboring Bases • Notation: A k = k th column of A • Suppose we have a feasible basis B (|B|=m, A B full rank) – It defines BFS x where x B =A B-1 b ¸ and x B =0 • Can we find a basis “similar” to B but containing some k B? • Suppose we increase x k from 0 to ² for some k B – We’ll violate the constraints Ax=b unless we modify x B x 1 x 2 Solutions of Ax=b Feasible region x 3 Example: Just one constraint: A = [1, 1, 1], b = [1] Feasible region: P = { x : x 1 +x 2 +x 3 =1, x ¸ 0 } BFS x =(1,0,0), basis B={1} Increase x 2 to ² . Infeasible! Modify x 1 to 1- ² . Feasible! Increase ² to 1. Get BFS y =(0,1,0). x (1, ² ,0) (1- ² , ² ,0) (0,1,0) How did we decide this? Neighboring Bases • Notation: A k = k th column of A • Suppose we have a feasible basis B (|B|=m, A B full rank) – It defines BFS x where x B =A B-1 b ¸ and x B =0 • Can we find a basis “similar” to B but containing some k B? • Suppose we increase x k from 0 to ² for some k B – We’ll violate the constraints Ax=b unless we modify x B – Replace x B with y B satisfying A B y B + ² A k =b – Given that y B [ {k} =0 and y k = ² , there is a unique y ensuring Ay=b A B y B + ² A k = b ) y B = A B-1 (b- ² A k ) = x B- ² A B-1 A k – So y( ² )=x+ ² d where: d B =-A B-1 A k , d k =1,...
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Lecture5 - C&O 355 Lecture 5 N Harvey...

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