On the Relationship between Conjugate Gradient and Optimal First-Order Methods for Convex Optimization

dc.contributor.authorKarimi, Sahar
dc.date.accessioned2014-01-23T20:23:30Z
dc.date.available2014-01-23T20:23:30Z
dc.date.issued2014-01-23
dc.date.submitted2014
dc.description.abstractIn a series of work initiated by Nemirovsky and Yudin, and later extended by Nesterov, first-order algorithms for unconstrained minimization with optimal theoretical complexity bound have been proposed. On the other hand, conjugate gradient algorithms as one of the widely used first-order techniques suffer from the lack of a finite complexity bound. In fact their performance can possibly be quite poor. This dissertation is partially on tightening the gap between these two classes of algorithms, namely the traditional conjugate gradient methods and optimal first-order techniques. We derive conditions under which conjugate gradient methods attain the same complexity bound as in Nemirovsky-Yudin's and Nesterov's methods. Moreover, we propose a conjugate gradient-type algorithm named CGSO, for Conjugate Gradient with Subspace Optimization, achieving the optimal complexity bound with the payoff of a little extra computational cost. We extend the theory of CGSO to convex problems with linear constraints. In particular we focus on solving $l_1$-regularized least square problem, often referred to as Basis Pursuit Denoising (BPDN) problem in the optimization community. BPDN arises in many practical fields including sparse signal recovery, machine learning, and statistics. Solving BPDN is fairly challenging because the size of the involved signals can be quite large; therefore first order methods are of particular interest for these problems. We propose a quasi-Newton proximal method for solving BPDN. Our numerical results suggest that our technique is computationally effective, and can compete favourably with the other state-of-the-art solvers.en
dc.identifier.urihttp://hdl.handle.net/10012/8189
dc.language.isoenen
dc.pendingfalse
dc.publisherUniversity of Waterlooen
dc.subjectConvex Optimizationen
dc.subjectFirst-Order Methodsen
dc.subjectConjugate Gradienten
dc.subjectProximal Quasi-Newton Methodsen
dc.subjectBasis Pursuit Denoising Problemen
dc.subjectL1-regularized Least Square problemen
dc.subject.programCombinatorics and Optimizationen
dc.titleOn the Relationship between Conjugate Gradient and Optimal First-Order Methods for Convex Optimizationen
dc.typeDoctoral Thesisen
uws-etd.degreeDoctor of Philosophyen
uws-etd.degree.departmentCombinatorics and Optimizationen
uws.peerReviewStatusUnrevieweden
uws.scholarLevelGraduateen
uws.typeOfResourceTexten

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