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Unformatted text preview: Solving Convex Programs by Random Walks DIMITRIS BERTSIMAS AND SANTOSH VEMPALA M.I.T., Cambridge, Massachusetts Abstract. Minimizing a convex function over a convex set in ndimensional space is a basic, general problem with many interesting special cases. Here, we present a simple new algorithm for convex optimization based on sampling by a random walk. It extends naturally to minimizing quasiconvex functions and to other generalizations. Categories and Subject Descriptors: F.2 [ Theory of Computation ]: Analysis of Algorithms and Problem Complexity; G.3 [ Probability and Statistics ]: stochastic processes General Terms: Algorithms, Theory Additional Key Words and Phrases: Convex programs, random walks, polynomial time 1 . Introduction The problem of minimizing a convex function over a convex set in R n is a common generalization of wellknown geometric optimization problems such as linear pro gramming as well as a variety of combinatorial optimization problems including matchings, flows and matroid intersection, all of which have polynomialtime algo rithms. As such, it represents a frontier of polynomialtime solvability and occupies a central place in the theory of algorithms. In his groundbreaking work, Khachiyan [1979] showed that the Ellipsoid method [Yudin and Nemirovski 1976] solves linear programs in polynomial time. Subse quently, Karp and Papadimitriou [1982], Padberg and Rao [1981], and Gr¨otschel et al. [1981] independently discovered the wide applicability of the Ellipsoid method to combinatorial optimization problems. This culminated in the book by the last set of authors [Gr¨otschel et al. 1988], in which it is shown that the Ellipsoid method solves the problem of minimizing a convex function over a convex set in R n spec ified by a separation oracle , that is, a procedure that given a point x , either reports D. Bertsimas was partially supported by the MITSingapore alliance. S. Vempala was supported by NSF CAREER award CCR9875024 and a Sloan Foundation Fellowship. Authors’ addresses: D. Bertsimas, Sloan School of Management and Operations Research Center, M.I.T., Cambridge, MA 02139; S. Vempala, Mathematics Department, M.I.T., Cambridge, MA 02139, email: { dbertsim,vempala } @mit.edu Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 1515 Broadway, New York, NY 10036 USA, fax: + 1 (212) 8690481, or [email protected] (212) 8690481, or [email protected]...
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This note was uploaded on 01/26/2010 for the course COMPUTER S 15853 taught by Professor Guyblelloch during the Fall '09 term at Carnegie Mellon.
 Fall '09
 GuyBlelloch
 Algorithms

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