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# lect6 - ISE 536Fall03 Linear Programming and Extensions...

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ISE 536–Fall03: Linear Programming and Extensions September 17, 2003 Lecture 6: Simplex Method, Moving in the Polyhedra Lecturer: Fernando Ord´o˜nez 1 Conceptual Algorithm to solve LP Consider the following to solve min c t x | x P , where P = { x ∈ < n | Ax = b, x 0 } . 1. Compute Cone( P ) if Cone( P ) = , then P is bounded goto 3. 2. Find w j extreme rays in Cone( P ). If c t w j < 0 then problem unbounded, STOP. 3. Find x i extreme points of P , move toward the one with smallest c t x i If A is m × n , How many extreme rays and extreme points do we have to ﬁnd in this algorithm?? 2 Feasible Directions and moving from a BFS Deﬁnition For a convex set P , and a point x P , a vector d is a feasible direction if x + θd P for some θ > 0. The simplex algorithm moves from BFS to adjacent BFS, that is solutions that have all but one of the same basic variables. 1

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Consider a BFS x = ( x B ,x N ) and a feasible direction d = ( d B ,d N ) such that the direction brings non-basic variable
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lect6 - ISE 536Fall03 Linear Programming and Extensions...

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