SIAM Journal on Imaging Sciences, ISSN 1936-4954, 08/2014, Volume 7, Issue 3, pp. 1588 - 1623

Alternating direction methods are a common tool for general mathematical programming and optimization. These methods have become particularly important in the...

Splitting | Accelerated | Method of multipliers | Bregman | Nesterov | ADMM | Optimization | splitting | MATHEMATICS, APPLIED | RECONSTRUCTION | SIGNAL RECOVERY | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | PROXIMAL POINT ALGORITHM | accelerated | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | VARIATIONAL-INEQUALITIES | COMPUTER SCIENCE, SOFTWARE ENGINEERING | DECONVOLUTION | MINIMIZATION | optimization | method of multipliers | SHRINKAGE | CONVERGENCE | HILBERT-SPACE | MONOTONE-OPERATORS | Algorithms | Image processing | Mathematical analysis | Tools | Minimization | Mathematical models | Convergence

Splitting | Accelerated | Method of multipliers | Bregman | Nesterov | ADMM | Optimization | splitting | MATHEMATICS, APPLIED | RECONSTRUCTION | SIGNAL RECOVERY | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | PROXIMAL POINT ALGORITHM | accelerated | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | VARIATIONAL-INEQUALITIES | COMPUTER SCIENCE, SOFTWARE ENGINEERING | DECONVOLUTION | MINIMIZATION | optimization | method of multipliers | SHRINKAGE | CONVERGENCE | HILBERT-SPACE | MONOTONE-OPERATORS | Algorithms | Image processing | Mathematical analysis | Tools | Minimization | Mathematical models | Convergence

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2012, Volume 50, Issue 2, pp. 700 - 709

Alternating direction methods (ADMs) have been well studied in the literature, and they have found many efficient applications in various fields. In this note,...

Mathematical inequalities | Algorithms | Approximation | Lagrangian function | Variational inequalities | Linearization | Alternating direction method | Convergence rate | Split inexact Uzawa method | Convex programming | VARIATIONAL-INEQUALITIES | MATHEMATICS, APPLIED | RECONSTRUCTION | alternating direction method | ALGORITHM | convex programming | SUM | convergence rate | variational inequalities | split inexact Uzawa method | MONOTONE-OPERATORS

Mathematical inequalities | Algorithms | Approximation | Lagrangian function | Variational inequalities | Linearization | Alternating direction method | Convergence rate | Split inexact Uzawa method | Convex programming | VARIATIONAL-INEQUALITIES | MATHEMATICS, APPLIED | RECONSTRUCTION | alternating direction method | ALGORITHM | convex programming | SUM | convergence rate | variational inequalities | split inexact Uzawa method | MONOTONE-OPERATORS

Journal Article

SIAM JOURNAL ON SCIENTIFIC COMPUTING, ISSN 1064-8275, 2011, Volume 33, Issue 1, pp. 250 - 278

In this paper, we propose and study the use of alternating direction algorithms for several l(1)-norm minimization problems arising from sparse solution...

BASIS PURSUIT | MATHEMATICS, APPLIED | IMAGE-RESTORATION | RECONSTRUCTION | primal | alternating direction method | SIGNAL RECOVERY | THRESHOLDING ALGORITHM | augmented Lagrangian function | dual | LINEAR INVERSE PROBLEMS | MINIMIZATION | SHRINKAGE | MONOTONE VARIATIONAL-INEQUALITIES | l-minimization | compressive sensing

BASIS PURSUIT | MATHEMATICS, APPLIED | IMAGE-RESTORATION | RECONSTRUCTION | primal | alternating direction method | SIGNAL RECOVERY | THRESHOLDING ALGORITHM | augmented Lagrangian function | dual | LINEAR INVERSE PROBLEMS | MINIMIZATION | SHRINKAGE | MONOTONE VARIATIONAL-INEQUALITIES | l-minimization | compressive sensing

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 2012, Volume 32, Issue 1, pp. 227 - 245

The matrix completion problem is to complete an unknown matrix from a small number of entries, and it captures many applications in diversified areas....

noise | convex programming | low rank | matrix completion | nuclear norm | alternating direction method | MATHEMATICS, APPLIED | MINIMIZATION | APPROXIMATION | MONOTONE VARIATIONAL-INEQUALITIES | ALGORITHM

noise | convex programming | low rank | matrix completion | nuclear norm | alternating direction method | MATHEMATICS, APPLIED | MINIMIZATION | APPROXIMATION | MONOTONE VARIATIONAL-INEQUALITIES | ALGORITHM

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 3/2016, Volume 66, Issue 3, pp. 889 - 916

The formulation $$\begin{aligned} \min _{x,y} ~f(x)+g(y),\quad \text{ subject } \text{ to } Ax+By=b, \end{aligned}$$ min x , y f ( x ) + g ( y ) , subject to A...

Alternating direction method of multipliers | Computational Mathematics and Numerical Analysis | Global convergence | Linear convergence | Algorithms | Strong convexity | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Mathematics | Distributed computing | MATHEMATICS, APPLIED | MINIMIZATION | Distributed computingb | PROXIMAL POINT ALGORITHM | OPTIMIZATION | SPLITTING ALGORITHMS | FLOW ALGORITHMS | MONOTONE-OPERATORS | Image processing | Methods | Machine learning | Splitting | Multipliers | Mathematical analysis | Mathematical models | Convexity | Statistics | Optimization | Convergence

Alternating direction method of multipliers | Computational Mathematics and Numerical Analysis | Global convergence | Linear convergence | Algorithms | Strong convexity | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Mathematics | Distributed computing | MATHEMATICS, APPLIED | MINIMIZATION | Distributed computingb | PROXIMAL POINT ALGORITHM | OPTIMIZATION | SPLITTING ALGORITHMS | FLOW ALGORITHMS | MONOTONE-OPERATORS | Image processing | Methods | Machine learning | Splitting | Multipliers | Mathematical analysis | Mathematical models | Convexity | Statistics | Optimization | Convergence

Journal Article

Information Processing Letters, ISSN 0020-0190, 04/2017, Volume 120, pp. 16 - 22

A monotone drawing of a graph is a straight-line planar drawing of such that every pair of vertices is connected by a path that is monotone with respect to...

Schnyder woods | Monotone drawings | Graph algorithms | Hamiltonian graphs | Maximal planar graphs | CONVEX DRAWINGS | DIMENSION | PLANAR GRAPHS | COMPUTER SCIENCE, INFORMATION SYSTEMS | Algorithms

Schnyder woods | Monotone drawings | Graph algorithms | Hamiltonian graphs | Maximal planar graphs | CONVEX DRAWINGS | DIMENSION | PLANAR GRAPHS | COMPUTER SCIENCE, INFORMATION SYSTEMS | Algorithms

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2012, Volume 22, Issue 2, pp. 313 - 340

We consider the linearly constrained separable convex minimization problem whose objective function is separable into m individual convex functions with...

Alternating direction method | Gaussian back substitution | Separable structure | Convex programming | SUPERRESOLUTION | SPARSITY | MATHEMATICS, APPLIED | alternating direction method | IMAGE-RECONSTRUCTION | DECOMPOSITION | convex programming | RANK | ALGORITHMS | separable structure | PENALTY | MONOTONE VARIATIONAL-INEQUALITIES

Alternating direction method | Gaussian back substitution | Separable structure | Convex programming | SUPERRESOLUTION | SPARSITY | MATHEMATICS, APPLIED | alternating direction method | IMAGE-RECONSTRUCTION | DECOMPOSITION | convex programming | RANK | ALGORITHMS | separable structure | PENALTY | MONOTONE VARIATIONAL-INEQUALITIES

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 01/2013, Volume 82, Issue 281, pp. 301 - 329

In this paper, we are interested in the application of the ALM and the ADM for some nuclear norm involved minimization problems. When the resulting subproblems...

Mathematical problems | Algorithms | Approximation | Objective functions | Machine learning | Linear transformations | Matrices | Lagrangian function | Linearization | Mathematical programming | Alternating direction method | Low-rank | Augmented Lagrangian method | Nuclear norm | Convex programming | Linearized | MATHEMATICS, APPLIED | augmented Lagrangian method | nuclear norm | IMAGE-RESTORATION | RECONSTRUCTION | alternating direction method | RANK | ALGORITHMS | BREGMAN METHOD | L-MINIMIZATION | MONOTONE VARIATIONAL-INEQUALITIES | low-rank | linearized | CONVERGENCE | OPTIMIZATION | REGULARIZATION

Mathematical problems | Algorithms | Approximation | Objective functions | Machine learning | Linear transformations | Matrices | Lagrangian function | Linearization | Mathematical programming | Alternating direction method | Low-rank | Augmented Lagrangian method | Nuclear norm | Convex programming | Linearized | MATHEMATICS, APPLIED | augmented Lagrangian method | nuclear norm | IMAGE-RESTORATION | RECONSTRUCTION | alternating direction method | RANK | ALGORITHMS | BREGMAN METHOD | L-MINIMIZATION | MONOTONE VARIATIONAL-INEQUALITIES | low-rank | linearized | CONVERGENCE | OPTIMIZATION | REGULARIZATION

Journal Article

Computational Optimization and Applications, ISSN 0926-6003, 9/2019, Volume 74, Issue 1, pp. 67 - 92

We revisit the classical Douglas–Rachford (DR) method for finding a zero of the sum of two maximal monotone operators. Since the practical performance of the...

Adaptive step-size | 65K05 | Maximal monotone inclusions | Mathematics | Statistics, general | Optimization | Douglas–Rachford method | 90C25 | Operations Research/Decision Theory | Convex and Discrete Geometry | Operations Research, Management Science | Non-stationary iteration | Alternating direction methods of multipliers | 47H05 | 65J15 | SPLITTING METHODS | MATHEMATICS, APPLIED | Douglas-Rachford method | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SHRINKAGE | ALGORITHMS | SELECTION | Algebra | Management science | Analysis | Methods | Direct reduction | Operators | Computational mathematics | Multipliers | Numerical methods | Convergence

Adaptive step-size | 65K05 | Maximal monotone inclusions | Mathematics | Statistics, general | Optimization | Douglas–Rachford method | 90C25 | Operations Research/Decision Theory | Convex and Discrete Geometry | Operations Research, Management Science | Non-stationary iteration | Alternating direction methods of multipliers | 47H05 | 65J15 | SPLITTING METHODS | MATHEMATICS, APPLIED | Douglas-Rachford method | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SHRINKAGE | ALGORITHMS | SELECTION | Algebra | Management science | Analysis | Methods | Direct reduction | Operators | Computational mathematics | Multipliers | Numerical methods | Convergence

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 5/2012, Volume 51, Issue 2, pp. 261 - 273

The covariance selection problem captures many applications in various fields, and it has been well studied in the literature. Recently, an l 1-norm penalized...

Log-likelihood | Alternating direction method | Computational Mathematics and Numerical Analysis | Algorithms | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | l 1 -norm | Mathematics | Covariance selection problem | MATHEMATICS, APPLIED | l-norm | MONOTONE VARIATIONAL-INEQUALITIES | OPTIMIZATION | Yuan (China) | Models | Analysis | Methods

Log-likelihood | Alternating direction method | Computational Mathematics and Numerical Analysis | Algorithms | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | l 1 -norm | Mathematics | Covariance selection problem | MATHEMATICS, APPLIED | l-norm | MONOTONE VARIATIONAL-INEQUALITIES | OPTIMIZATION | Yuan (China) | Models | Analysis | Methods

Journal Article

Optimization Methods and Software, ISSN 1055-6788, 03/2014, Volume 29, Issue 2, pp. 239 - 263

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...

matrix separation | augmented Lagrangian function | alternating direction method | Alternating direction method | Augmented Lagrangian function | Matrix separation | MATHEMATICS, APPLIED | RECONSTRUCTION | THRESHOLDING ALGORITHM | DECOMPOSITION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | MONOTONE VARIATIONAL-INEQUALITIES | COMPLETION | ROBUST UNCERTAINTY PRINCIPLES | Operations research | Lagrange multiplier | Algorithms | Optimization

matrix separation | augmented Lagrangian function | alternating direction method | Alternating direction method | Augmented Lagrangian function | Matrix separation | MATHEMATICS, APPLIED | RECONSTRUCTION | THRESHOLDING ALGORITHM | DECOMPOSITION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | MONOTONE VARIATIONAL-INEQUALITIES | COMPLETION | ROBUST UNCERTAINTY PRINCIPLES | Operations research | Lagrange multiplier | Algorithms | Optimization

Journal Article

SIAM Journal on Imaging Sciences, ISSN 1936-4954, 03/2015, Volume 8, Issue 1, pp. 644 - 681

We present a novel framework, namely, accelerated alternating direction method of multipliers (AADMM), for acceleration of linearized ADMM. The basic idea of...

Alternating direction method of multipliers | Accelerated gradient method | Convex optimization | MATHEMATICS, APPLIED | accelerated gradient method | PENALTY SCHEMES | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | PROXIMAL POINT ALGORITHM | CONVEX-OPTIMIZATION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | VARIATION IMAGE-RECONSTRUCTION | VARIATIONAL-INEQUALITIES | COMPUTER SCIENCE, SOFTWARE ENGINEERING | PRIMAL-DUAL ALGORITHMS | ITERATION-COMPLEXITY | alternating direction method of multipliers | convex optimization | FORWARD-BACKWARD | SADDLE-POINT | MONOTONE-OPERATORS | Multipliers | Algorithms | Saddle points | Imaging | Constants | Acceleration | Optimization | Convergence

Alternating direction method of multipliers | Accelerated gradient method | Convex optimization | MATHEMATICS, APPLIED | accelerated gradient method | PENALTY SCHEMES | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | PROXIMAL POINT ALGORITHM | CONVEX-OPTIMIZATION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | VARIATION IMAGE-RECONSTRUCTION | VARIATIONAL-INEQUALITIES | COMPUTER SCIENCE, SOFTWARE ENGINEERING | PRIMAL-DUAL ALGORITHMS | ITERATION-COMPLEXITY | alternating direction method of multipliers | convex optimization | FORWARD-BACKWARD | SADDLE-POINT | MONOTONE-OPERATORS | Multipliers | Algorithms | Saddle points | Imaging | Constants | Acceleration | Optimization | Convergence

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 01/2019, Volume 346, pp. 237 - 246

In this paper, monotonicity-preserving interpolation is generalized to direction-consistent interpolation. The conditions for constructing direction-consistent...

Curve representation | Shape-preserving | Hermite interpolation | Tangent direction | MATHEMATICS, APPLIED | ENERGY | MONOTONIC DATA | CUBIC-SPLINES | SPLINE INTERPOLATION

Curve representation | Shape-preserving | Hermite interpolation | Tangent direction | MATHEMATICS, APPLIED | ENERGY | MONOTONIC DATA | CUBIC-SPLINES | SPLINE INTERPOLATION

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2013, Volume 23, Issue 4, pp. 2183 - 2207

We introduce a novel matrix recurrence yielding a new spectral analysis of the local transient convergence behavior of the alternating direction method of...

Quadratic programming | Linear programming | ADMM | MATHEMATICS, APPLIED | PENALTY | SHRINKAGE | MONOTONE VARIATIONAL-INEQUALITIES | LASSO | linear programming | SPLITTING ALGORITHMS | CONVEX-OPTIMIZATION | quadratic programming | OPERATORS | INVERSE

Quadratic programming | Linear programming | ADMM | MATHEMATICS, APPLIED | PENALTY | SHRINKAGE | MONOTONE VARIATIONAL-INEQUALITIES | LASSO | linear programming | SPLITTING ALGORITHMS | CONVEX-OPTIMIZATION | quadratic programming | OPERATORS | INVERSE

Journal Article

SIAM Journal on Optimization, ISSN 1052-6234, 2013, Volume 23, Issue 1, pp. 475 - 507

In this paper, we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point-to-set maximal monotone operator with...

Proximal | Extragradient | Decomposition | Monotone operator | Inclusion problem | Complexity | complexity | MATHEMATICS, APPLIED | MAXIMAL MONOTONE-OPERATORS | inclusion problem | decomposition | PROXIMAL POINT ALGORITHM | CONVEX-OPTIMIZATION | monotone operator | FAMILY | VARIATIONAL-INEQUALITIES | extragradient | PROGRAMS | ENLARGEMENTS | proximal | SADDLE-POINT | PROJECTIVE SPLITTING METHODS | Operators | Multipliers | Algorithms | Approximation | Programming | Blocking | Ergodic processes | Convergence

Proximal | Extragradient | Decomposition | Monotone operator | Inclusion problem | Complexity | complexity | MATHEMATICS, APPLIED | MAXIMAL MONOTONE-OPERATORS | inclusion problem | decomposition | PROXIMAL POINT ALGORITHM | CONVEX-OPTIMIZATION | monotone operator | FAMILY | VARIATIONAL-INEQUALITIES | extragradient | PROGRAMS | ENLARGEMENTS | proximal | SADDLE-POINT | PROJECTIVE SPLITTING METHODS | Operators | Multipliers | Algorithms | Approximation | Programming | Blocking | Ergodic processes | Convergence

Journal Article

European Journal of Operational Research, ISSN 0377-2217, 06/2019

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 1/2015, Volume 164, Issue 1, pp. 218 - 233

We consider combining the generalized alternating direction method of multipliers, proposed by Eckstein and Bertsekas, with the logarithmic–quadratic proximal...

Generalized alternating direction method of multipliers | 65K05 | Mathematics | Theory of Computation | Optimization | Calculus of Variations and Optimal Control; Optimization | Variational inequality | 90C25 | Operations Research/Decision Theory | 90C33 | Convergence rate | Applications of Mathematics | Engineering, general | Logarithmic–quadratic proximal method | VARIATIONAL-INEQUALITIES | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | DECOMPOSITION METHODS | Logarithmic-quadratic proximal method | POINT ALGORITHM | MONOTONE-OPERATORS | Yuan (China) | Algorithms | Analysis | Methods | Studies | Mathematical analysis | Convergence | Multipliers | Inequalities | Regularization | Ergodic processes | Complexity

Generalized alternating direction method of multipliers | 65K05 | Mathematics | Theory of Computation | Optimization | Calculus of Variations and Optimal Control; Optimization | Variational inequality | 90C25 | Operations Research/Decision Theory | 90C33 | Convergence rate | Applications of Mathematics | Engineering, general | Logarithmic–quadratic proximal method | VARIATIONAL-INEQUALITIES | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | DECOMPOSITION METHODS | Logarithmic-quadratic proximal method | POINT ALGORITHM | MONOTONE-OPERATORS | Yuan (China) | Algorithms | Analysis | Methods | Studies | Mathematical analysis | Convergence | Multipliers | Inequalities | Regularization | Ergodic processes | Complexity

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 2014, Volume 226, pp. 367 - 373

The alternating direction method of multipliers (ADMM) is known to be a classic and efficient method for constrained optimization problem with two blocks of...

Convergence rate | Alternating direction method of multipliers | Convex programming | Variational inequalities | MATHEMATICS, APPLIED | MONOTONE VARIATIONAL-INEQUALITIES | ALGORITHM | DECOMPOSITION

Convergence rate | Alternating direction method of multipliers | Convex programming | Variational inequalities | MATHEMATICS, APPLIED | MONOTONE VARIATIONAL-INEQUALITIES | ALGORITHM | DECOMPOSITION

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 8/2019, Volume 182, Issue 2, pp. 640 - 666

This paper proposes a partially inexact proximal alternating direction method of multipliers for computing approximate solutions of a linearly constrained...

Ergodic iteration-complexity | Alternating direction method of multipliers | Mathematics | Theory of Computation | Optimization | Pointwise iteration-complexity | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operations Research/Decision Theory | Convex program | Relative error criterion | Hybrid extragradient method | 65K10 | 49M27 | Applications of Mathematics | Engineering, general | 47H05 | 90C60 | MATHEMATICS, APPLIED | TUMOR | CLASSIFICATION | ENLARGEMENT | REGULARIZED ADMM | CANCER | PREDICTION | EXTRAGRADIENT | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE RATE | MONOTONE-OPERATORS | POINT ALGORITHM | Computational geometry | Error analysis | Multipliers | Convexity | Iterative methods | Complexity

Ergodic iteration-complexity | Alternating direction method of multipliers | Mathematics | Theory of Computation | Optimization | Pointwise iteration-complexity | Calculus of Variations and Optimal Control; Optimization | 90C25 | Operations Research/Decision Theory | Convex program | Relative error criterion | Hybrid extragradient method | 65K10 | 49M27 | Applications of Mathematics | Engineering, general | 47H05 | 90C60 | MATHEMATICS, APPLIED | TUMOR | CLASSIFICATION | ENLARGEMENT | REGULARIZED ADMM | CANCER | PREDICTION | EXTRAGRADIENT | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE RATE | MONOTONE-OPERATORS | POINT ALGORITHM | Computational geometry | Error analysis | Multipliers | Convexity | Iterative methods | Complexity

Journal Article

Minimax Theory and its Applications, ISSN 2199-1413, 2016, Volume 1, Issue 1, pp. 29 - 49

Journal Article

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