lecture9 - infeasible yes no e62 lecture 9 Reduced Profts...

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e62: lecture 9 13/10/10 Simplex Algorithm 1 c e62: lecture 9 13/10/10 Simplex Algorithm 2 simplex algorithm problem data initial vertex ( A, b, c ) v today’s version assumes existence of an optimal solution max cx s . t . Ax = b x 0 standard form LP x * optimal vertex e62: lecture 9 13/10/10 Vertices Vertex = intersection of N linearly independent constraint boundaries M come from Ax = b N ! M are nonnegativity constraints At least N ! M components are zero N ! M nonbasic variables M basic variables Example N =5, M =3, B = {1, 4, 5} Solving for a vertex given B 3 max cx s . t . Ax = b x 0 } } M constraints N constraints B B 0 1.5 3 4.5 6 1 2 3 4 5 Ax = b x B =0 A B x B = b x B =0 x B = A - 1 B b x B =0 e62: lecture 9 13/10/10 Edges Def. adjacent vertices: share N-1 constraint boundaries An edge is a part of the line that satisFes these constraints Swap one constraint to switch between adjacent vertices In a symmetric form LP, swap one nonbasic variable 4 0 1.5 3 4.5 6 1 2 3 4 5 0 1.5 3 4.5 6 1 2 3 4 5 adjacent
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e62: lecture 9 13/10/10 Traversing an Edge Example N =5, M =3, B = {1, 4, 5} 5 0 1.5 3 4.5 6 1 2 3 4 5 feasible to increase ? pick nonbasic variable increase until
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Unformatted text preview: infeasible yes no e62: lecture 9 13/10/10 Reduced Profts • Def. reduced proFt = objective value increase per unit nonbasic variable increase 6 1.5 3 4.5 6 1 2 3 4 5 x-→ x + Δ cx-→ cx + c Δ reduced proft = c Δ Δ 2 e62: lecture 9 13/10/10 Algorithm and Convergence • Termination • At an optimal solution, no positive reduced profts • Finite time convergence • There are a fnite number o± vertices 7 1.5 3 4.5 6 1 2 3 4 5 compute reduced costs traverse maximizing edge maximum positive ? yes no e62: lecture 9 13/10/10 Degeneracy and Cycling • Def. degeneracy = when there is a zero-valued basic variable • Example where N =5, M =3 (3 basic variables) • Intersection o± N+1 constraint boundaries • Def. cycling = keep swapping basic variables without changing solution 8 1.5 3 4.5 6 1 2 3 4 5...
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This note was uploaded on 07/29/2011 for the course MS&E 111 at Stanford.

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lecture9 - infeasible yes no e62 lecture 9 Reduced Profts...

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