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Unformatted text preview: Advanced Operations Research Techniques IE316 Lecture 3 Dr. Ted Ralphs IE316 Lecture 3 1 Reading for This Lecture Bertsimas 2.12.2 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. 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 halfspace . Notes : IE316 Lecture 3 4 Convex Sets Definition 4. A set S R n is convex if x,y S and R with 1 , we have x + (1 ) y S . Definition 5. Let x 1 ,...,x k R n and R k + be given such that T 1 = 1 . The vector k i =1 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 : 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 ....
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This note was uploaded on 08/06/2008 for the course IE 316 taught by Professor Ralphs during the Fall '08 term at Lehigh University .
 Fall '08
 Ralphs
 Operations Research

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