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Solving Convex Programs by Random Walks

Solving Convex Programs by Random Walks - Solving Convex...

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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 n -dimensional 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 quasi-convex 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 well-known 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 polynomial-time algo- rithms. As such, it represents a frontier of polynomial-time 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 etal.[1981]independentlydiscoveredthewideapplicabilityoftheEllipsoidmethod 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 MIT-Singapore alliance. S. Vempala was supported by NSF CAREER award CCR-9875024 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, e-mail: { 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) 869-0481, or [email protected]
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