A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
Randomized algorithms
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
Location of the Maximum on Unimodal Surfaces
Journal of the ACM (JACM)
Solving convex programs by random walks
Journal of the ACM (JACM)
The D-decomposition technique for linear matrix inequalities
Automation and Remote Control
Approximating the centroid is hard
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
A probabilistic analytic center cutting plane method for feasibility of uncertain LMIs
Automatica (Journal of IFAC)
Random walks on polytopes and an affine interior point method for linear programming
Proceedings of the forty-first annual ACM symposium on Theory of computing
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
Randomized Algorithms for Analysis and Control of Uncertain Systems: With Applications
Randomized Algorithms for Analysis and Control of Uncertain Systems: With Applications
Survey paper: Research on probabilistic methods for control system design
Automatica (Journal of IFAC)
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We propose a randomized method for general convex optimization problems; namely, the minimization of a linear function over a convex body. The idea is to generate $N$ random points inside the body, choose the best one, and cut the part of the body defined by the linear constraint. We analyze the convergence properties of the algorithm from a theoretical viewpoint, i.e., under the rather standard assumption that an algorithm for uniform generation of random points in the convex body is available. Under this assumption, the expected rate of convergence for such method is proved to be geometric. This analysis is based on new results on the statistical properties of the empirical minimum over a convex body that we obtained in this paper. Moreover, explicit sample size results on convergence are derived. In particular, we compute the minimum number of random points that should be generated at each step in order to guarantee that, in a probabilistic sense, the method performs better than the deterministic center-of-gravity algorithm. From a practical viewpoint, we show how the method can be implemented using hit-and-run versions of Markov-chain Monte Carlo algorithms and exemplify the performance of this implementable modification via a number of illustrative problems. A crucial notion for the hit-and-run implementation is that of boundary oracle, which is available for most optimization problems including linear matrix inequalities and many other kinds of constraints. Preliminary numerical results for semidefinite programs are presented confirming that the randomized approach might be competitive to modern deterministic convex optimization methods.