Studies integrating geometry, probability, and optimization under convexity

Convexity has played a major role in a variety of fields over the past decades. Nevertheless, the convexity assumption continues to reveal new theoretical paradigms and applications. This dissertation explores convexity in the intersection of three fields, namely, geometry, probability, and optimization. We study in depth a variety of geometric quantities. These quantities are used to describe the behavior of different algorithms. In addition, we investigate how to algorithmically manipulate these geometric quantities. This leads to algorithms capable of transforming ill-behaved instances into well-behaved ones. In particular, we provide probabilistic methods that carry out such task efficiently by exploiting the geometry of the problem. More specific contributions of this dissertation are as follows. (i) We conduct a broad exploration of the symmetry function of convex sets and propose efficient methods for its computation in the polyhedral case. (ii) We also relate the symmetry function with the computational complexity of an interior-point method to solve a homogeneous conic system. (iii) Moreover, we develop a family of pre-conditioners based on the symmetry function and projective transformations for such interior-point method.

Department dc:contributor.department

Massachusetts Institute of Technology. Operations Research Center.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2006

Author dc:creator

• Nogueira, Alexandre Belloni

Advisor dc:contributor.advisor

• Robert M. Freund.

• Operations Research Center.

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/1721.1/36227

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/36227