Lecture4

# Lecture4 - Introduction to Mathematical Programming IE406...

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Unformatted text preview: Introduction to Mathematical Programming IE406 Lecture 4 Dr. Ted Ralphs IE406 Lecture 4 1 Reading for This Lecture • Bertsimas 2.2-2.4 IE406 Lecture 4 2 The Two Crude Petroleum Example Revisited • Recall the Two Crude Petroleum example. • We showed graphically that the optimal solution was an extreme point . • How did we figure out the coordinates of the optimal point? IE406 Lecture 4 3 Binding Constraints Consider a polyhedron P = { x ∈ R n | Ax ≥ b } . Definition 1. If a vector ˆ x satisfies a i ˆ x = b i , then we say the corresponding constraint is binding at ˆ x . Theorem 1. Let ˆ x ∈ R n be given and let I = { i | a i ˆ x = b i } represent the set of constraints that are binding at ˆ x . Then the following are equivalent: • There exist n vectors in the set { a i | i ∈ I } that are linearly independent. • The span of the vectors { a i | i ∈ I } is R n . • The system of equations a i x = b i , i ∈ I, x ∈ R n has the unique solution ˆ x . If the vectors { a j | j ∈ J } for some J ⊆ [1 , m ] are linearly independent, we will say that the corresponding constraints are also linearly independent. IE406 Lecture 4 4 Basic Solutions...
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## This note was uploaded on 08/06/2008 for the course IE 406 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture4 - Introduction to Mathematical Programming IE406...

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