Lecture5

# Lecture5 - Introduction to Mathematical Programming IE406...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Introduction to Mathematical Programming IE406 Lecture 5 Dr. Ted Ralphs IE406 Lecture 5 1 Reading for This Lecture • Bertsimas 2.5-2.7 IE406 Lecture 5 2 Existence of Extreme Points Definition 1. A polyhedron P ∈ R n contains a line if there exists a vector x ∈ P and a nonzero vector d ∈ R n such that x + λd ∈ P ∀ λ ∈ R . Theorem 1. Suppose that the polyhedron P = { x ∈ R n | Ax ≥ b } is nonempty. Then the following are equivalent: • The polyhedron P has at least one extreme point. • The polyhedron P does not contain a line. • There exist n rows of A that are linearly independent. IE406 Lecture 5 3 Optimality of Extreme Points Theorem 2. Let P ⊆ R n be a polyhedron and consider the problem min x ∈P c x for a given c ∈ R n . If P has at least one extreme point and there exists an optimal solution, then there exists an optimal solution that is an extreme point....
View Full Document

## 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 .

### Page1 / 11

Lecture5 - Introduction to Mathematical Programming IE406...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online