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Unformatted text preview: The Ellipsoid Algorithm for Linear Programming Lecturer: Sanjeev Arora, COS 521, Fall 2005 Princeton University Scribe Notes: Siddhartha Brahma The Ellipsoid algorithm for linear programming is a specific application of the ellipsoid method developed by Soviet mathematicians Shor(1970), Yudin and Nemirovskii(1975). Khachiyan(1979) applied the ellipsoid method to derive the first polynomial time algorithm for linear programming. Although the algorithm is theoretically better than the Simplex algorithm, which has an exponential running time in the worst case, it is very slow practically and not competitive with Simplex. Nevertheless, it is a very important theoretical tool for developing polynomial time algorithms for a large class of convex op- timization problems, which are much more general than linear programming. We will start of with a few definitions and then consider the actual algorithm. We will consider general linear programs of the following form defined on vectors in R n . maximize c T x Ax b x where A is a m n real constraint matrix and x,c R n . Definition 1. A Hyperplane is defined to be the set of points satisfying the linear equation ax = b , where a,x,b R n . Definition 2. A Convex Set K R n is a set of points such that x,y K , x +(1- ) y K , where [0 , 1]. A Convex Body is a closed and bounded convex set. A few examples of convex sets and bodies are as follows: 1. Hypercube length l is the set of all x such that 0 x i l, 1 i n . 2. Ball of radius r around the origin is the set of all x such that n X i =1 x 2 i r 2 . 3. An axis aligned ellipsoid is the set of all x such that n X i =1 x 2 i /a 2 i 1 where a i s are nonzero reals....
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This note was uploaded on 10/23/2011 for the course CS 7520 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
- Spring '08