Date of Award

Summer 8-2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics & Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

Yuesheng Xu

Committee Member

Yaohang Li

Committee Member

Ke Shi

Committee Member

Guohui Song

Committee Member

Lucia Tabacu

Abstract

The aim of this dissertation is to develop efficient inexact fixed-point proximity algorithms with convergence guaranteed for nonsmooth convex optimization problems encountered in data science. Nonsmooth convex optimization is one of the core methodologies in data science to acquire knowledge from real-world data and has wide applications in various fields, including signal/image processing, machine learning and distributed computing. In particular, in the context of image reconstruction, compressed sensing and sparse machine learning, either the objective functions or the constraints of the modeling optimization problems are nondifferentiable. Hence, traditional methods such as the gradient descent method and the Newton method are not applicable since gradients of the objective functions or the constraints do not exist. Fixed-point proximity algorithms were developed via subdifferentials of the objective function to address the challenges. The theory of nonexpansive averaged operators was successfully employed in the existing analysis of exact/inexact fixed-point proximity algorithms for nonsmooth convex optimization. However, this framework has imposed restricted constraints on the algorithm formulation, which slows down the convergence and conceals relations between different algorithms.

In this work, we characterize the solutions of convex optimization as fixed-points of certain operators, and then adopt the matrix splitting technique to obtain a framework of fully implicit fixed-point proximity algorithms. This results in a new class of quasiaveraged operators, which extends the class of nonexpansive averaged operators. Such framework covers and generalizes most of the existing popular algorithms for nonsmooth convex optimization. To deal with the implicitness of this framework, we follow the inspiration of the Schur’s lemma on the uniform boundedness of infinite matrices and propose a framework of inexact fixed-point iterations of quasiaveraged operators. This framework generalizes the inexact iterations of nonexpansive averaged operators. A combination of the frameworks of inexact fixed-point iterations and the implicit fixed-point proximity algorithms leads to the framework of inexact fixed-point proximity algorithms, which further extends existing methods for nonsmooth convex optimization. Numerical experiments on image deblurring problems demonstrate the advantages of inexact fixed-point proximity algorithms over existing explicit algorithms.

DOI

10.25777/adyv-ra74

ISBN

9798351481449

ORCID

0000-0001-9930-5215

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