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Unformatted text preview: Advanced Operations Research Techniques IE316 Lecture 6 Dr. Ted Ralphs IE316 Lecture 6 1 Reading for This Lecture Bertsimas 3.13.2. IE316 Lecture 6 2 Summary of What Weve Learned So Far We are interested in the extreme points of polyhedra. There is a onetoone correspondence between the extreme points of a polyhedron and the basic feasible solutions . We can construct basic solutions by Choosing a basis B of m linearly independent columns of A . Solve the system Bx B = b to obtain the values of the basic variables. Set x N = 0 . We can move between adjacent (nondegenerate) basic solutions by removing one column of the basis and replacing it with another. In the presence of degeneracy , we might stay at the same extreme point. These are the building blocks we need to construct algorithms for solving LPs. IE316 Lecture 6 3 Iterative Search Algorithms Many optimization algorithms are iterative in nature. Geometrically, this means that they move from a given starting point to a new point in a specified search direction . This search direction is calculated to be both feasible and improving . The process stops when we can no longer find a feasible, improving direction. For linear programs, it is always possible to find a feasible improving direction if we are not at an optimal point. This is essentially what makes linear programs easy to solve. IE316 Lecture 6 4 Feasible and Improving Directions Definition 1. Let x be an element of a polyhedron P . A vector d R n is said to be a feasible direction if there exists R + such that x + d P ....
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This note was uploaded on 08/06/2008 for the course IE 316 taught by Professor Ralphs during the Fall '08 term at Lehigh University .
 Fall '08
 Ralphs
 Operations Research

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