This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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 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 : 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 ....
View
Full
Document
 Fall '08
 Ralphs
 Linear Algebra, Operations Research, Vector Space, Dr. Ted Ralphs

Click to edit the document details