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 x satisfies a i x = b i , then we say the corresponding constraint is binding at x . Theorem 1. Let x R n be given and let I = { i | a i x = b i } represent the set of constraints that are binding at 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 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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Lecture4 - Introduction to Mathematical Programming IE406...

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