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Lecture3 - Advanced Operations Research Techniques IE316...

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Advanced Operations Research Techniques IE316 Lecture 3 Dr. Ted Ralphs
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IE316 Lecture 3 1 Reading for This Lecture Bertsimas 2.1-2.2
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IE316 Lecture 3 2 From Last Time Recall the Two Crude Petroleum example. In the example, the optimal solution was a “ corner point .” We saw that the following are possible outcomes of solving an optimization problem: In fact, we will see that these are the only possibilities . We will also see that when there is an optimal solution and at least one “corner point,” there is an optimal solution that is a “corner point.”
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IE316 Lecture 3 3 Some Definitions Definition 1. A polyhedron is a set of the form { x R n | Ax b } , where A R m × n and b R m . Definition 2. A set S R n is bounded if there exists a constant K such that | x i | < K x S, i [1 , n ] . Definition 3. Let a R n and b R be given. The set { x R n | a T x = b } is called a hyperplane . The set { x R n | a T x b } is called a half-space . Notes :
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IE316 Lecture 3 4 Convex Sets Definition 4. A set S R n is convex if x, y S and λ R with 0 λ 1 , we have λx + (1 - λ ) y S . Definition 5. Let x 1 , . . . , x k R n and λ R k + be given such that λ T 1 = 1 . The vector k i =1 λ i x i is said to be a convex combination of x 1 , . . . , x k . The convex hull of x 1 , . . . , x k is the set of all convex combinations of these vectors. Notes :
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IE316 Lecture 3 5 Properties of Convex Sets The following properties can be derived from the definitions: The intersection of convex sets is convex . Every polyhedron is a convex set . The convex combination of a finite number of elements of a convex set also belongs to the set . The convex hull of a finite number of vectors is a convex set .
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