Lecture3 - Introduction to Mathematical Programming IE406...

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Unformatted text preview: Introduction to Mathematical Programming IE406 Lecture 3 Dr. Ted Ralphs IE406 Lecture 3 1 Reading for This Lecture • Bertsimas 2.1-2.2 IE406 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.” IE406 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 x = b } is called a hyperplane . • The set { x ∈ R n | a x ≥ b } is called a half-space . Notes : IE406 Lecture 3 4 Convex Sets Definition 4. A set S ⊆ R n is convex if ∀ x, y ∈ S and λ ∈ R with ≤ 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 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 : IE406 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 406 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture3 - Introduction to Mathematical Programming IE406...

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