A Unified Framework for Sparse Relaxed Regularized Regression: SR3

Regularized regression problems are ubiquitous in statistical modeling, signal processing, and machine learning. Sparse regression in particular has been instrumental in scientific model discovery, including compressed sensing applications, vari- able selection, and high-dimensional analysis. We propose a broad framework for sparse relaxed regularized regression, called SR3. The key idea is to solve a relaxation of … Read more

A TVSCAD approach for image deblurring with impulsive noise

We consider image deblurring problem in the presence of impulsive noise. It is known that \emph{total variation} (TV) regularization with L1-norm penalized data fitting (TVL1 for short) works reasonably well only when the level of impulsive noise is relatively low. For high level impulsive noise, TVL1 works poorly. The reason is that all data, both … Read more

A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions

We analyze alternating descent algorithms for minimizing the sum of a quadratic function and block separable non-smooth functions. In case the quadratic interactions between the blocks are pairwise, we show that the schemes can be accelerated, leading to improved convergence rates with respect to related accelerated parallel proximal descent. As an application we obtain very … Read more

A Preconditioner for a Primal-Dual Newton Conjugate Gradients Method for Compressed Sensing Problems

In this paper we are concerned with the solution of Compressed Sensing (CS) problems where the signals to be recovered are sparse in coherent and redundant dictionaries. We extend a primal-dual Newton Conjugate Gradients (pdNCG) method for CS problems. We provide an inexpensive and provably effective preconditioning technique for linear systems using pdNCG. Numerical results … Read more

Fast Bundle-Level Type Methods for unconstrained and ball-constrained convex optimization

It has been shown in \cite{Lan13-1} that the accelerated prox-level (APL) method and its variant, the uniform smoothing level (USL) method, have optimal iteration complexity for solving black-box and structured convex programming problems without requiring the input of any smoothness information. However, these algorithms require the assumption on the boundedness of the feasible set and … Read more

Inertial primal-dual algorithms for structured convex optimization

The primal-dual algorithm recently proposed by Chambolle \& Pock (abbreviated as CPA) for structured convex optimization is very efficient and popular. It was shown by Chambolle \& Pock in \cite{CP11} and also by Shefi \& Teboulle in \cite{ST14} that CPA and variants are closely related to preconditioned versions of the popular alternating direction method of … Read more

Self Equivalence of the Alternating Direction Method of Multipliers

The alternating direction method of multipliers (ADM or ADMM) breaks a complex optimization problem into much simpler subproblems. The ADM algorithms are typically short and easy to implement yet exhibit (nearly) state-of-the-art performance for large-scale optimization problems. To apply ADM, we first formulate a given problem into the “ADM-ready” form, so the final algorithm depends … Read more

A Second-Order Method for Compressed Sensing Problems with Coherent and Redundant Dictionaries

In this paper we are interested in the solution of Compressed Sensing (CS) problems where the signals to be recovered are sparse in coherent and redundant dictionaries. CS problems of this type are convex with non-smooth and non-separable regularization term, therefore a specialized solver is required. We propose a primal-dual Newton Conjugate Gradients (pdNCG) method. … Read more

An Efficient Augmented Lagrangian Method with Applications to Total Variation Minimization

Based on the classic augmented Lagrangian multiplier method, we propose, analyze and test an algorithm for solving a class of equality-constrained non-smooth optimization problems (chiefly but not necessarily convex programs) with a particular structure. The algorithm effectively combines an alternating direction technique with a nonmonotone line search to minimize the augmented Lagrangian function at each … Read more

Superiorization: An optimization heuristic for medical physics

Purpose: To describe and mathematically validate the superiorization methodology, which is a recently-developed heuristic approach to optimization, and to discuss its applicability to medical physics problem formulations that specify the desired solution (of physically given or otherwise obtained constraints) by an optimization criterion. Methods: The superiorization methodology is presented as a heuristic solver for a … Read more