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 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
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 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
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 Fall '09
 GuyBlelloch
 Algorithms, Convex set, Convex function, SANTOSH VEMPALA

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