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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 22 Dr. Ted Ralphs IE417 Lecture 22 1 Reading for This Lecture Sections 10.110.2 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 * + d S, (0 , ) , > } 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 * + d ) < f ( x * ) (0 , ) , > } 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....
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 Fall '08
 Linderoth

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