Many problems of recent interest in statistics and machine learning can be posed in the
framework of convex optimization. Due to the explosion in size and complexity of modern …

This monograph is about a class of optimization algorithms called proximal algorithms. Much
like Newton's method is a standard tool for solving unconstrained smooth optimization …

The convex feasibility problem (CFP) is to find a feasible point in the intersection of finitely
many convex and closed sets. If the intersection is empty then the CFP is inconsistent and a …

We introduce a first-order method for solving very large convex cone programs. The method
uses an operator splitting method, the alternating directions method of multipliers, to solve …

This tutorial paper presents the basic notation and results of monotone operators and
operator splitting methods, with a focus on convex optimization. A very wide variety of …

The proximity operator of a convex function is a natural extension of the notion of a
projection operator onto a convex set. This tool, which plays a central role in the analysis …

Iterative methods for finding fixed points of non-expansive operators in Hilbert spaces have
been described in many publications. In this monograph we try to present the methods in a …

The Variational Inequality Problem [6, 52, 118, 119] has been and will continue to be one of
the central problems in nonlinear analysis and is defined as follows: given monotone …

The phase retrieval problem is of paramount importance in various areas of applied physics
and engineering. The state of the art for solving this problem in two dimensions relies …

We report on progress in algorithms for iterative phase retrieval. The theory of convex
optimization is used to develop and to gain insight into counterparts for the nonconvex …