Lecture22 - Advanced Mathematical Programming IE417 Lecture...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 22 Dr. Ted Ralphs IE417 Lecture 22 1 Reading for This Lecture • Sections 10.1-10.2 IE417 Lecture 22 2 Methods of Feasible Directions • In Chapter 9, we looked at methods of using unconstrained optimization techniques on constrained problems. • These methods enforced some of the constraints implicitly . • Now, we look at the methods that explicitly enforce feasibility while ensuring convergence. • Recall the concept of an improving, feasible direction . IE417 Lecture 22 3 Feasible and Improving Directions Definition 1. Let S be a nonempty set in R n and let x * ∈ clS . The cone of feasible directions of S at x * is given by D = { d : d 6 = 0 and x * + λd ∈ S, ∀ λ ∈ (0 ,δ ) , ∃ δ > } Definition 2. Let S be a nonempty set in R n and let x * ∈ clS . Given a function f : R n → R , the cone of improving directions of f at x * is given by F = { d : f ( x * + λd ) < f ( x * ) ∀ λ ∈ (0 ,δ ) , ∃ δ > } IE417 Lecture 22 4 The Case of Linear Constraints • Consider the problem min f ( x ) s . t . Ax ≤ b Cx = d • The direction d is a feasible direction from the point x * if and only if A 1 d ≤ and Cd = 0 where A 1 are the constraints binding at x * . • If 5 f ( x * ) > d < , then d is an improving direction. IE417 Lecture 22 5 Generating Improving Feasible Directions (Linear Case) • We want to generate a direction for which – 5 f ( x * ) > d < – A 1 d ≤ – Cd = 0 • Idea : Solve the following optimization problem min 5 f ( x * ) > d s . t . A 1 d ≤ Cd = 0 • We also need a normalizing constraint. IE417 Lecture 22 6 Remember Way Back When...Remember Way Back When....
View Full Document

This note was uploaded on 08/06/2008 for the course IE 417 taught by Professor Linderoth during the Fall '08 term at Lehigh University .

Page1 / 19

Lecture22 - Advanced Mathematical Programming IE417 Lecture...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online