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Mixed-integer convex optimization : outer approximation algorithms and modeling power

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dc.contributor.advisor Juan Pablo Vielma. en_US Lubin, Miles (Miles C.) en_US
dc.contributor.other Massachusetts Institute of Technology. Operations Research Center. en_US 2018-02-08T15:57:33Z 2018-02-08T15:57:33Z 2017 en_US 2017 en_US
dc.description Thesis: Ph. D., Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2017. en_US
dc.description This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. en_US
dc.description Cataloged from student-submitted PDF version of thesis. en_US
dc.description Includes bibliographical references (pages 137-143). en_US
dc.description.abstract In this thesis, we study mixed-integer convex optimization, or mixed-integer convex programming (MICP), the class of optimization problems where one seeks to minimize a convex objective function subject to convex constraints and integrality restrictions on a subset of the variables. We focus on two broad and complementary questions on MICP. The first question we address is, "what are efficient methods for solving MICP problems?" The methodology we develop is based on outer approximation, which allows us, for example, to reduce MICP to a sequence of mixed-integer linear programming (MILP) problems. By viewing MICP from the conic perspective of modern convex optimization as defined by Ben-Tal and Nemirovski, we obtain significant computational advances over the state of the art, e.g., by automating extended formulations by using disciplined convex programming. We develop the first finite-time outer approximation methods for problems in general mixed-integer conic form (which includes mixed-integer second-order-cone programming and mixed-integer semidefinite programming) and implement them in an open-source solver, Pajarito, obtaining competitive performance with the state of the art. The second question we address is, "which nonconvex constraints can be modeled with MICP?" This question is important for understanding both the modeling power gained in generalizing from MILP to MICP and the potential applicability of MICP to nonconvex optimization problems that may not be naturally represented with integer variables. Among our contributions, we completely characterize the case where the number of integer assignments is bounded (e.g., mixed-binary), and to address the more general case we develop the concept of "rationally unbounded" convex sets. We show that under this natural restriction, the projections of MICP feasible sets are well behaved and can be completely characterized in some settings. en_US
dc.description.statementofresponsibility by Miles Lubin. en_US
dc.format.extent 143 pages en_US
dc.language.iso eng en_US
dc.publisher Massachusetts Institute of Technology en_US
dc.rights MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission. en_US
dc.rights.uri en_US
dc.subject Operations Research Center. en_US
dc.title Mixed-integer convex optimization : outer approximation algorithms and modeling power en_US
dc.title.alternative Outer approximation algorithms and modeling power 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 1020068050 en_US

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