Tractable stochastic analysis in high dimensions via robust optimization

• Main Author: Bandi, Chaithanya
• Other Authors: Dimitris Bertsimas.
• Format: Others
• Language: English
• Published: Massachusetts Institute of Technology 2013
• Subjects: Operations Research Center.
• Online Access: http://hdl.handle.net/1721.1/82870

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2013. === Cataloged from PDF version of thesis. === Includes bibliographical references (pages 201-207). === Modern probability theory, whose foundation is based on the axioms set forth by Kolmogorov, is currently the major tool for performance analysis in stochastic systems. While it offers insights in understanding such systems, probability theory, in contrast to optimization, has not been developed with computational tractability as an objective when the dimension increases. Correspondingly, some of its major areas of application remain unsolved when the underlying systems become multidimensional: Queueing networks, auction design in multi-item, multi-bidder auctions, network information theory, pricing multi-dimensional financial contracts, among others. We propose a new approach to analyze stochastic systems based on robust optimization. The key idea is to replace the Kolmogorov axioms and the concept of random variables as primitives of probability theory, with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem. In addition, we observe that several desired system properties such as incentive compatibility and individual rationality in auction design and correct decoding in information theory are naturally expressed in the language of robust optimization. In this way, the performance analysis questions become highly structured optimization problems (linear, semidefinite, mixed integer) for which there exist efficient, practical algorithms that are capable of solving problems in high dimensions. We demonstrate that the proposed approach achieves computationally tractable methods for (a) analyzing queueing networks (Chapter 2) (b) designing multi-item, multi-bidder auctions with budget constraints, (Chapter 3) (c) characterizing the capacity region and designing optimal coding and decoding methods in multi-sender, multi-receiver communication channels (Chapter 4). === by Chaithanya Bandi. === Ph.D.