Zaiwen Wen

Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm

The matrix completion problem is to recover a low-rank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclear-norm minimization which requires computing singular value decompositions—a task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving large-scale problems, we propose a low-rank factorization model and construct a nonlinear successive over-relaxation (SOR) algorithm that only requires solving a linear least squares problem per iteration. Extensive numerical experiments show that the algorithm can reliably solve a wide range of problems at a speed at least several times faster than many nuclear-norm minimization algorithms. In addition, convergence of this nonlinear SOR algorithm to a stationary point is analyzed.

Alternating direction augmented Lagrangian methods for semidefinite programming

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.

Augmented Lagrangian alternating direction method for matrix separation based on low-rank factorization

The matrix separation problem aims to separate a low-rank matrix and a sparse matrix from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications. Nuclear-norm minimization models have been proposed for matrix separation and proved to yield exact separations under suitable conditions. These models, however, typically require the calculation of a full or partial singular value decomposition at every iteration that can become increasingly costly as matrix dimensions and rank grow. To improve scalability, in this paper, we propose and investigate an alternative approach based on solving a non-convex, low-rank factorization model by an augmented Lagrangian alternating direction method. Numerical studies indicate that the effectiveness of the proposed model is limited to problems where the sparse matrix does not dominate the low-rank one in magnitude, though this limitation can be alleviated by certain data pre-processing techniques. On the other hand, extensive numerical results show that, within its applicability range, the proposed method in general has a much faster solution speed than nuclear-norm minimization algorithms and often provides better recoverability.

A Fast Algorithm for Sparse Reconstruction Based on Shrinkage, Subspace Optimization, and Continuation

We propose a fast algorithm for solving the

An alternating direction algorithm for matrix completion with nonnegative factors

\ell_1

Trust, But Verify: Fast and Accurate Signal Recovery From 1-Bit Compressive Measurements

-regularized minimization problem

Alternating direction methods for classical and ptychographic phase retrieval

\min_{x\in\mathbb{R}^n}\mu\|x\|_1+\|Ax-b\|^2_2

Decentralized low-rank matrix completion

for recovering sparse solutions to an undetermined system of linear equations

Block Coordinate Descent Methods for Semidefinite Programming

Ax=b

On the convergence of an active-set method for ℓ1 minimization

. The algorithm is divided into two stages that are performed repeatedly. In the first stage a first-order iterative “shrinkage” method yields an estimate of the subset of components of