Enhancements and computational evaluation of the hit-and-run random walk on polyhedra

The symmetry function of a convex set offers us numerous useful information about the set in relation to probabilistic theory and geometric properties. The symmetry function is a measure of how symmetric the convex set is, and for a point, intuitively it measures how symmetric the set is with respect to that point. We call a point of high symmetry value a deep point. A random walk is a procedure that starts from a particular point in Rn and at each iteration, moves to a "neighboring" point according to some probability distribution that depends solely on the current point. The Hit-and-Run random walk on a convex set S picks a random line e through the point, and at next iteration goes to a new point that is chosen uniformly on the chord ℓ [intersection] S. In this thesis, we analyze and investigate the effectiveness of the Hit-and-Run random walk to compute a deep point in a convex body, given a randomly generated convex set. The effectiveness is evaluated in terms of the role of the starting point and the likelihood that the random walk will enter the zone of high symmetry. Additionally, some known probabilistic properties of the symmetry function are tested using the random walk, from which the integrity of the code is also verified.

Department dc:contributor.department

Massachusetts Institute of Technology. Computation for Design and Optimization Program.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2006

Author dc:creator

• Liang, Jiajie, S.M. Massachusetts Institute of Technology

Advisor dc:contributor.advisor

• Robert M. Freund.

• Computation for Design and Optimization Program.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/39216

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/39216