Lecture7

# Lecture7 - Introduction to Mathematical Programming IE406...

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Introduction to Mathematical Programming IE406 Lecture 7 Dr. Ted Ralphs

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IE406 Lecture 7 1 Reading for This Lecture Bertsimas 3.2-3.4
IE406 Lecture 7 2 The Simplex Method A typical iteration of the simplex method: 1. Start with a speciﬁed basis matrix B and a corresponding BFS x 0 . 2. Compute the reduced cost vector ¯ c . If ¯ c 0 , then x 0 is optimal . 3. Otherwise, choose j for which ¯ c j < 0 . 4. Compute u = B - 1 A j . If u 0 , then θ * = and the LP is unbounded . 5. Otherwise, θ * = min { i | u i > 0 } x 0 B ( i ) u i . 6. Choose l such that θ * = x 0 B ( l ) u l and form a new basis matrix, replacing A B ( l ) with A j . 7. The values of the new basic variables are x 1 j = θ * and x 1 B ( i ) = x 0 B ( i ) - θ * u i if i ± = l .

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IE406 Lecture 7 3 Some Notes on the Simplex Method We will see later how to construct an initial basic feasible solution. We saw last time that each iteration of the simplex methods ends with a new basic feasible solution (assuming nondegeneracy). This is all we need to prove the following result: Theorem 1. Consider a linear program over a nonempty polyhedron P and assume every basic feasible solution is nondegenerate . Then the simplex method terminates after a ﬁnite number of iterations in one of the following two conditions: We obtain an optimal basis and a corresponding optimal basic feasible solution. We obtain a vector d R n such that Ad = 0 , d 0 , and c ± d < 0 , and the LP is unbounded .
IE406 Lecture 7 4 Pivot Selection The process of removing one variable and replacing from the basis and replacing it with another is called pivoting . We have the freedom to choose the entering variable from among a list of candidates. How do we make this choice?

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Lecture7 - Introduction to Mathematical Programming IE406...

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