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

In 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

AMODEL OF INTERNET SHOPPER BEHAVIOR, A CROSS SECTOR ANALYSIS

This research aims to enhance our knowledge of Internet purchase behavior by proposing a new model which draws from three disciplines: consumer behavior, decision analysis and IS. A research methodology has been designed to capture the dynamic process that Internet shoppers follow when they are engaged in a shopping experience and the influences of their interactions with the environment on their behavior. Data from an Internet panel data provider, video recording sessions and questionnaires will be collected to capture the purchase behavior process and establish the contextual factors and the possibilities of cross-channel behavior in the three different sectors of banking, groceries and mobile phones. The proposed model will be revised based on the results of our study. The theoretical and empirical results will inform the three literatures of this study. Companies will also be able to devise winning strategies by better understanding their consumers’ Internet shopping behavior