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Robust optimization, game theory, and variational inequalities

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dc.contributor.advisor Dimitris Bertsimas. en_US Aghassi, Michele Leslie en_US
dc.contributor.other Massachusetts Institute of Technology. Operations Research Center. en_US 2006-07-31T15:22:05Z 2006-07-31T15:22:05Z 2005 en_US 2005 en_US
dc.description Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2005. en_US
dc.description Includes bibliographical references (p. 193-109). en_US
dc.description.abstract We propose a robust optimization approach to analyzing three distinct classes of problems related to the notion of equilibrium: the nominal variational inequality (VI) problem over a polyhedron, the finite game under payoff uncertainty, and the network design problem under demand uncertainty. In the first part of the thesis, we demonstrate that the nominal VI problem is in fact a special instance of a robust constraint. Using this insight and duality-based proof techniques from robust optimization, we reformulate the VI problem over a polyhedron as a single- level (and many-times continuously differentiable) optimization problem. This reformulation applies even if the associated cost function has an asymmetric Jacobian matrix. We give sufficient conditions for the convexity of this reformulation and thereby identify a class of VIs, of which monotone affine (and possibly asymmetric) VIs are a special case, which may be solved using widely-available and commercial-grade convex optimization software. In the second part of the thesis, we propose a distribution-free model of incomplete- information games, in which the players use a robust optimization approach to contend with payoff uncertainty. en_US
dc.description.abstract (cont.) Our "robust game" model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative, distribution-free equilibrium concept, for which, in contrast to ex post equilibria, existence is guaranteed. We show that computation of "robust-optimization equilibria" is analogous to that of Nash equilibria of complete- information games. Our results cover incomplete-information games either involving or not involving private information. In the third part of the thesis, we consider uncertainty on the part of a mechanism designer. Specifically, we present a novel, robust optimization model of the network design problem (NDP) under demand uncertainty and congestion effects, and under either system- optimal or user-optimal routing. We propose a corresponding branch and bound algorithm which comprises the first constructive use of the price of anarchy concept. In addition, we characterize conditions under which the robust NDP reduces to a less computationally demanding problem, either a nominal counterpart or a single-level quadratic optimization problem. Finally, we present a novel traffic "paradox," illustrating counterintuitive behavior of changes in cost relative to changes in demand. en_US
dc.description.statementofresponsibility by Michele Leslie Aghassi. en_US
dc.format.extent 209 p. en_US
dc.format.extent 10614705 bytes
dc.format.extent 10623510 bytes
dc.format.mimetype application/pdf
dc.format.mimetype application/pdf
dc.language.iso eng en_US
dc.publisher Massachusetts Institute of Technology en_US
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. en_US
dc.subject Operations Research Center. en_US
dc.title Robust optimization, game theory, and variational inequalities en_US
dc.type Thesis en_US Ph.D. en_US
dc.contributor.department Massachusetts Institute of Technology. Operations Research Center. en_US
dc.identifier.oclc 64565159 en_US

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