Faster Stochastic Alternating Direction Method of Multipliers for Nonconvex Optimization
University of Pittsburgh（匹兹堡大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；MIIT Key Laboratory of Pattern Analysis & Machine Intelligence（工业和信息化部模式分析与机器智能重点实验室）；JD Finance America Corporation（京东金融美国公司）
Feihu Huang, Songcan Chen, Heng Huang
备注：
Published in ICML 2019, 43 pages. arXiv admin note: text overlap with arXiv:1907.13463
In this paper, we propose a faster stochastic alternating direction method of multipliers (ADMM) for nonconvex optimization by using a new stochastic path-integrated differential estimator (SPIDER), called as SPIDER-ADMM. Moreover, we prove that the SPIDER-ADMM achieves a record-breaking incremental first-order oracle (IFO) complexity of $\mathcal{O}(n+n^{1/2}ε^{-1})$ for finding an $ε$-approximate stationary point, which improves the deterministic ADMM by a factor $\mathcal{O}(n^{1/2})$, where $n$ denotes the sample size. As one of major contribution of this paper, we provide a new theoretical analysis framework for nonconvex stochastic ADMM methods with providing the optimal IFO complexity. Based on this new analysis framework, we study the unsolved optimal IFO complexity of the existing non-convex SVRG-ADMM and SAGA-ADMM methods, and prove they have the optimal IFO complexity of $\mathcal{O}(n+n^{2/3}ε^{-1})$. Thus, the SPIDER-ADMM improves the existing stochastic ADMM methods by a factor of $\mathcal{O}(n^{1/6})$. Moreover, we extend SPIDER-ADMM to the online setting, and propose a faster online SPIDER-ADMM. Our theoretical analysis shows that the online SPIDER-ADMM has the IFO complexity of $\mathcal{O}(ε^{-\frac{3}{2}})$, which improves the existing best results by a factor of $\mathcal{O}(ε^{-\frac{1}{2}})$. Finally, the experimental results on benchmark datasets validate that the proposed algorithms have faster convergence rate than the existing ADMM algorithms for nonconvex optimization.
Alternating gradient-descent-ascent (AltGDA) is an optimization algorithm that has been widely used for model training in various machine learning applications, which aims to solve a nonconvex minimax optimization problem. However, the existing studies show that it suffers from a high computation complexity in nonconvex minimax optimization. In this paper, we develop a single-loop and fast AltGDA-type algorithm that leverages proximal gradient updates and momentum acceleration to solve regularized nonconvex minimax optimization problems. By leveraging the momentum acceleration technique, we pr…
Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problems
Shanghai University（上海大学）；Newtouch Center for Mathematics of Shanghai University（上海大学纽崔克数学中心）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
The mirror descent algorithm (MDA) generalizes gradient descent by using a Bregman divergence to replace squared Euclidean distance. In this paper, we similarly generalize the alternating direction method of multipliers (ADMM) to Bregman ADMM (BADMM), which allows the choice of different Bregman divergences to exploit the structure of problems. BADMM provides a unified framework for ADMM and its variants, including generalized ADMM, inexact ADMM and Bethe ADMM. We establish the global convergence and the $O(1/T)$ iteration complexity for BADMM. In some cases, BADMM can be faster than ADMM by a…
Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA（赫尔曼·米勒工程与系统工程学院，佐治亚理工学院，美国亚特兰大，GA）；Sauder School of Business, University of British Columbia, Vancouver, BC, Canada（萨德勒商学院，不列颠哥伦比亚大学，加拿大温哥华，BC）
We present a systematic weight pruning framework of deep neural networks (DNNs) using the alternating direction method of multipliers (ADMM). We first formulate the weight pruning problem of DNNs as a constrained nonconvex optimization problem, and then adopt the ADMM framework for systematic weight pruning. We show that ADMM is highly suitable for weight pruning due to the computational efficiency it offers. We achieve a much higher compression ratio compared with prior work while maintaining the same test accuracy, together with a faster convergence rate. Our models are released at https://g…
National University of Singapore（新加坡国立大学）；Peking University（北京大学）；Shanghai Jiao Tong University（上海交通大学）；AI Institute（360人工智能研究院）
Spectral Clustering (SC) is a widely used data clustering method which first learns a low-dimensional embedding $U$ of data by computing the eigenvectors of the normalized Laplacian matrix, and then performs k-means on $U^\top$ to get the final clustering result. The Sparse Spectral Clustering (SSC) method extends SC with a sparse regularization on $UU^\top$ by using the block diagonal structure prior of $UU^\top$ in the ideal case. However, encouraging $UU^\top$ to be sparse leads to a heavily nonconvex problem which is challenging to solve and the work (Lu, Yan, and Lin 2016) proposes a conv…
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Regularization is a popular technique in machine learning for model estimation and avoiding overfitting. Prior studies have found that modern ordered regularization can be more effective in handling highly correlated, high-dimensional data than traditional regularization. The reason stems from the fact that the ordered regularization can reject irrelevant variables and yield an accurate estimation of the parameters. How to scale up the ordered regularization problems when facing the large-scale training data remains an unanswered question. This paper explores the problem of parameter estimatio…
Sun Yat-Sen University（中山大学）；University of Science and Technology of China（中国科学技术大学）；Pazhou Lab（琶洲实验室）
This paper aims to solve a distributed learning problem under Byzantine attacks. In the underlying distributed system, a number of unknown but malicious workers (termed as Byzantine workers) can send arbitrary messages to the master and bias the learning process, due to data corruptions, computation errors or malicious attacks. Prior work has considered a total variation (TV) norm-penalized approximation formulation to handle the Byzantine attacks, where the TV norm penalty forces the regular workers' local variables to be close, and meanwhile, tolerates the outliers sent by the Byzantine work…
University of Michigan（密歇根大学）；IBM Research（IBM研究院）；Northwestern Polytechnical University, China（西北工业大学）
In this paper, we design and analyze a new zeroth-order online algorithm, namely, the zeroth-order online alternating direction method of multipliers (ZOO-ADMM), which enjoys dual advantages of being gradient-free operation and employing the ADMM to accommodate complex structured regularizers. Compared to the first-order gradient-based online algorithm, we show that ZOO-ADMM requires $\sqrt{m}$ times more iterations, leading to a convergence rate of $O(\sqrt{m}/\sqrt{T})$, where $m$ is the number of optimization variables, and $T$ is the number of iterations. To accelerate ZOO-ADMM, we propose…
Alternating Iteratively Reweighted $\ell_1$ and Subspace Newton Algorithms for Nonconvex Sparse Optimization
School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）；School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）；Center for Applied Mathematics of Henan Province, Henan University（河南大学河南省应用数学中心）
University of Southern California（南加州大学）；Indian Institute of Technology Kharagpur（印度克勒格布尔理工学院）；Facebook AI（Facebook人工智能实验室）
Quantization of the parameters of machine learning models, such as deep neural networks, requires solving constrained optimization problems, where the constraint set is formed by the Cartesian product of many simple discrete sets. For such optimization problems, we study the performance of the Alternating Direction Method of Multipliers for Quantization ($\texttt{ADMM-Q}$) algorithm, which is a variant of the widely-used ADMM method applied to our discrete optimization problem. We establish the convergence of the iterates of $\texttt{ADMM-Q}$ to certain $\textit{stationary points}$. To the bes…
The Alternating Direction Method of Multipliers (ADMM) has been studied for years. The traditional ADMM algorithm needs to compute, at each iteration, an (empirical) expected loss function on all training examples, resulting in a computational complexity proportional to the number of training examples. To reduce the time complexity, stochastic ADMM algorithms were proposed to replace the expected function with a random loss function associated with one uniformly drawn example plus a Bregman divergence. The Bregman divergence, however, is derived from a simple second order proximal function, th…
Typically, nonlinear Support Vector Machines (SVMs) produce significantly higher classification quality when compared to linear ones but, at the same time, their computational complexity is prohibitive for large-scale datasets: this drawback is essentially related to the necessity to store and manipulate large, dense and unstructured kernel matrices. Despite the fact that at the core of training a SVM there is a \textit{simple} convex optimization problem, the presence of kernel matrices is responsible for dramatic performance reduction, making SVMs unworkably slow for large problems. Aiming t…
Huawei Belgium Research Center（华为比利时研究中心）；Carnegie Mellon University（卡内基梅隆大学）；Université de Mons（蒙斯大学）
In this paper, we propose an algorithmic framework, dubbed inertial alternating direction methods of multipliers (iADMM), for solving a class of nonconvex nonsmooth multiblock composite optimization problems with linear constraints. Our framework employs the general minimization-majorization (MM) principle to update each block of variables so as to not only unify the convergence analysis of previous ADMM that use specific surrogate functions in the MM step, but also lead to new efficient ADMM schemes. To the best of our knowledge, in the nonconvex nonsmooth setting, ADMM used in combination wi…
In this work, we present a hardware compatible neural network training algorithm in which we used alternating direction method of multipliers (ADMM) and iterative least-square methods. The motive behind this approach was to conduct a method of training neural networks that is scalable and can be parallelised. These characteristics make this algorithm suitable for hardware implementation. We have achieved 6.9\% and 6.8\% better accuracy comparing to SGD and Adam respectively, with a four-layer neural network with hidden size of 28 on HIGGS dataset. Likewise, we could observe 21.0\% and 2.2\% ac…
University of Tennessee（田纳西大学）；Sun Yat-Sen University（中山大学）
Channel pruning has been identified as an effective approach to constructing efficient network structures. Its typical pipeline requires iterative pruning and fine-tuning. In this work, we propose a novel single-shot channel pruning approach based on alternating direction methods of multipliers (ADMM), which can eliminate the need for complex iterative pruning and fine-tuning procedure and achieve a target compression ratio with only one run of pruning and fine-tuning. To the best of our knowledge, this is the first study of single-shot channel pruning. The proposed method introduces filter-le…
Stochastic alternating direction method of multipliers (ADMM), which visits only one sample or a mini-batch of samples each time, has recently been proved to achieve better performance than batch ADMM. However, most stochastic methods can only achieve a convergence rate $O(1/\sqrt T)$ on general convex problems,where T is the number of iterations. Hence, these methods are not scalable with respect to convergence rate (computation cost). There exists only one stochastic method, called SA-ADMM, which can achieve convergence rate $O(1/T)$ on general convex problems. However, an extra memory is ne…
Syracuse University（雪城大学）；Florida International University（佛罗里达国际大学）；Northeastern University（东北大学）
Weight pruning methods for deep neural networks (DNNs) have been investigated recently, but prior work in this area is mainly heuristic, iterative pruning, thereby lacking guarantees on the weight reduction ratio and convergence time. To mitigate these limitations, we present a systematic weight pruning framework of DNNs using the alternating direction method of multipliers (ADMM). We first formulate the weight pruning problem of DNNs as a nonconvex optimization problem with combinatorial constraints specifying the sparsity requirements, and then adopt the ADMM framework for systematic weight …
In the paper, we study the stochastic alternating direction method of multipliers (ADMM) for the nonconvex optimizations, and propose three classes of the nonconvex stochastic ADMM with variance reduction, based on different reduced variance stochastic gradients. Specifically, the first class called the nonconvex stochastic variance reduced gradient ADMM (SVRG-ADMM), uses a multi-stage scheme to progressively reduce the variance of stochastic gradients. The second is the nonconvex stochastic average gradient ADMM (SAG-ADMM), which additionally uses the old gradients estimated in the previous i…
We analyze the performance of alternating minimization for loss functions optimized over two variables, where each variable may be restricted to lie in some potentially nonconvex constraint set. This type of setting arises naturally in high-dimensional statistics and signal processing, where the variables often reflect different structures or components within the signals being considered. Our analysis relies on the notion of local concavity coefficients, which has been proposed in Barber and Ha to measure and quantify the concavity of a general nonconvex set. Our results further reveal import…
This work investigates the theoretical performance of the alternating-direction method of multipliers (ADMM) as it applies to nonconvex optimization problems, and in particular, problems with nonconvex constraint sets. The alternating direction method of multipliers is an optimization method that has largely been analyzed for convex problems. The ultimate goal is to assess what kind of theoretical convergence properties the method has in the nonconvex case, and to this end, theoretical contributions are two-fold. First, this work analyzes the method with local solution of the ADMM subproblems,…
We introduce a generalization of the linearized Alternating Direction Method of Multipliers to optimize a real-valued function $f$ of multiple arguments with potentially multiple constraints $g_\circ$ on each of them. The function $f$ may be nonconvex as long as it is convex in every argument, while the constraints $g_\circ$ need to be convex but not smooth. If $f$ is smooth, the proposed Block-Simultaneous Direction Method of Multipliers (bSDMM) can be interpreted as a proximal analog to inexact coordinate descent methods under constraints. Unlike alternative approaches for joint solvers of m…
Iteratively Linearized Reweighted Alternating Direction Method of Multipliers for a Class of Nonconvex Problems
Department of Mathematics, National University of Defense Technology（国防科技大学数学系）；College of Computer, National University of Defense Technology（国防科技大学计算机学院）；The State Key Laboratory for High Performance Computation, National University of Defense Technology（国防科技大学高性能计算国家重点实验室）；Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University（湖南计算与模拟科学工程重点实验室，湘潭大学数学与计算科学学院）
This paper addresses a class of constrained optimization problems over networks in which local cost functions and constraints can be nonconvex. We propose an asynchronous distributed optimization algorithm, relying on the centralized Method of Multipliers, in which each node wakes up in an uncoordinated fashion and performs either a descent step on a local Augmented Lagrangian or an ascent step on the local multiplier vector. These two phases are regulated by a distributed logic-AND, which allows nodes to understand when the descent on the (whole) Augmented Lagrangian is sufficiently small. We…
We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear convergence. When the two sets are semi-algebraic and bounded, but not necessarily transversal, we nonetheless prove subsequence convergence.
The classic Alternating Direction Method of Multipliers (ADMM) is a popular framework to solve linear-equality constrained problems. In this paper, we extend the ADMM naturally to nonlinear equality-constrained problems, called neADMM. The difficulty of neADMM is to solve nonconvex subproblems. We provide globally optimal solutions to them in two important applications. Experiments on synthetic and real-world datasets demonstrate excellent performance and scalability of our proposed neADMM over existing state-of-the-start methods.
Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems
Shanghai University（上海大学）；Newtouch Center for Mathematics of Shanghai University（上海大学新触数学中心）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various convex or nonconvex composite problems that arise in the fields of image & signal processing and machine learning. In convex settings, numerous convergence results have been established for ADMM as well as its varieties. However, due to the absence of convexity, the convergence analysis of nonconvex ADMM is generally very difficult. In this paper we study the Bregman modification of ADMM (BADMM), which includes the conventional ADMM as a special case and often leads…
The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems, and show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. Fo…
University of Pittsburgh（匹兹堡大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；JD Finance America Corporation（京东金融美国公司）
Alternating direction method of multipliers (ADMM) is a popular optimization tool for the composite and constrained problems in machine learning. However, in many machine learning problems such as black-box attacks and bandit feedback, ADMM could fail because the explicit gradients of these problems are difficult or infeasible to obtain. Zeroth-order (gradient-free) methods can effectively solve these problems due to that the objective function values are only required in the optimization. Recently, though there exist a few zeroth-order ADMM methods, they build on the convexity of objective fu…
For some typical and widely used non-convex half-quadratic regularization models and the Ambrosio-Tortorelli approximate Mumford-Shah model, based on the Kurdyka-Łojasiewicz analysis and the recent nonconvex proximal algorithms, we developed an efficient preconditioned framework aiming at the linear subproblems that appeared in the nonlinear alternating minimization procedure. Solving large-scale linear subproblems is always important and challenging for lots of alternating minimization algorithms. By cooperating the efficient and classical preconditioned iterations into the nonlinear and nonc…
Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex sets is typically intractable, but we show that computationally-cheap inexact projections may suffice instead. In particular, if one set is defined by sufficiently regular smooth constraints, then projecting onto the approximation obtained by linearizing those constraints around the current iterate suffices. On the other hand, if one set is a smooth manifold …
National University of Defense Technology（国防科技大学）；Universidad de Zaragoza（萨拉戈萨大学）
In image processing, Total Variation (TV) regularization models are commonly used to recover blurred images. One of the most efficient and popular methods to solve the convex TV problem is the Alternating Direction Method of Multipliers (ADMM) algorithm, recently extended using the inertial proximal point method. Although all the classical studies focus on only a convex formulation, recent articles are paying increasing attention to the nonconvex methodology due to its good numerical performance and properties. In this paper, we propose to extend the classical formulation with a novel nonconve…
In this paper we study nonconvex and nonsmooth optimization problems with semi-algebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the Proximal Alternating Linearized Minimization (PALM) algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sp…
We propose two numerical algorithms in the fully nonconvex setting for the minimization of the sum of a smooth function and the composition of a nonsmooth function with a linear operator. The iterative schemes are formulated in the spirit of the proximal alternating direction method of multipliers and its linearized variant, respectively. The proximal terms are introduced via variable metrics, a fact which allows us to derive new proximal splitting algorithms for nonconvex structured optimization problems, as particular instances of the general schemes. Under mild conditions on the sequence of…
Shanghai University（上海大学）；Georgia Institute of Technology（佐治亚理工学院）
Much recent research effort has been directed to the development of efficient algorithms for solving minimax problems with theoretical convergence guarantees due to the relevance of these problems to a few emergent applications. In this paper, we propose a unified single-loop alternating gradient projection (AGP) algorithm for solving smooth nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. AGP employs simple gradient projection steps for updating the primal and dual variables alternatively at each iteration. We show that it can find an $\varepsilon$-stationary po…
Nonconvex and structured optimization problems arise in many engineering applications that demand scalable and distributed solution methods. The study of the convergence properties of these methods is in general difficult due to the nonconvexity of the problem. In this paper, two distributed solution methods that combine the fast convergence properties of augmented Lagrangian-based methods with the separability properties of alternating optimization are investigated. The first method is adapted from the classic quadratic penalty function method and is called the Alternating Direction Penalty M…
This paper presents a fully asynchronous and distributed approach for tackling optimization problems in which both the objective function and the constraints may be nonconvex. In the considered network setting each node is active upon triggering of a local timer and has access only to a portion of the objective function and to a subset of the constraints. In the proposed technique, based on the method of multipliers, each node performs, when it wakes up, either a descent step on a local augmented Lagrangian or an ascent step on the local multiplier vector. Nodes realize when to switch from the…
Zeroth-Order primal-dual Alternating Projection Gradient Algorithms for Nonconvex Minimax Problems with Coupled linear Constraints
Department of Mathematics, College of Sciences, Shanghai University（上海大学数学系）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Newtouch Center for Mathematics of Shanghai University（上海大学数学中心）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
This paper establishes the iteration-complexity of a Jacobi-type non-Euclidean proximal alternating direction method of multipliers (ADMM) for solving multi-block linearly constrained nonconvex programs. The subproblems of this ADMM variant can be solved in parallel and hence the method has great potential to solve large scale multi-block linearly constrained nonconvex programs. Moreover, our analysis allows the Lagrange multiplier to be updated with a relaxation parameter in the interval (0, 2).
Consider the minimization of a nonconvex differentiable function over a polyhedron. A popular primal-dual first-order method for this problem is to perform a gradient projection iteration for the augmented Lagrangian function and then update the dual multiplier vector using the constraint residual. However, numerical examples show that this approach can exhibit "oscillation" and may not converge. In this paper, we propose a proximal alternating direction method of multipliers for the multi-block version of this problem. A distinctive feature of this method is the introduction of a "smoothed" (…
In this paper we propose a proximal algorithm for minimizing an objective function of two block variables consisting of three terms: 1) a smooth function, 2) a nonsmooth function which is a composition between a strictly increasing, concave, differentiable function and a convex nonsmooth function, and 3) a smooth function which couples the two block variables. We propose a variable metric composite proximal alternating linearized minimization (CPALM) to solve this class of problems. Building on the powerful Kurdyka-Łojasiewicz property, we derive the convergence analysis and establish that eac…
In this paper, we study a general optimization model, which covers a large class of existing models for many applications in imaging sciences. To solve the resulting possibly nonconvex, nonsmooth and non-Lipschitz optimization problem, we adapt the alternating direction method of multipliers (ADMM) with a general dual step-size to solve a reformulation that contains three blocks of variables, and analyze its convergence. We show that for any dual step-size less than the golden ratio, there exists a computable threshold such that if the penalty parameter is chosen above such a threshold and the…
A lift-and-permute scheme of alternating direction method of multipliers (ADMM) is proposed for linearly constrained convex programming. It contains not only the newly developed balanced augmented Lagrangian method and its dual-primal variation, but also the proximal ADMM and Douglas-Rachford splitting algorithm. It helps to propose accelerated algorithms with worst-case $O(1/k^2)$ convergence rates in the case that the objective function to be minimized is strongly convex.
We propose a variant of the classical conditional gradient method for sparse inverse problems with differentiable measurement models. Such models arise in many practical problems including superresolution, time-series modeling, and matrix completion. Our algorithm combines nonconvex and convex optimization techniques: we propose global conditional gradient steps alternating with nonconvex local search exploiting the differentiable measurement model. This hybridization gives the theoretical global optimality guarantees and stopping conditions of convex optimization along with the performance an…
This technical note is an ancillary material for our research paper (Itoh and Parente, 2019). We discuss an alternating direction method of multipliers with generalized augmented terms (ADMM-GAT) and introduce a generalized residual balance technique for efficiently employing ADMM-GAT. These techniques are applied to least absolute deviation and its constrained version and their algorithmic details are presented. These algorithms are used for the implementation of the method described in (Itoh and Parente, 2019).
The support vector machine (SVM) was originally designed for binary classifications. A lot of effort has been put to generalize the binary SVM to multiclass SVM (MSVM) which are more complex problems. Initially, MSVMs were solved by considering their dual formulations which are quadratic programs and can be solved by standard second-order methods. However, the duals of MSVMs with regularizers are usually more difficult to formulate and computationally very expensive to solve. This paper focuses on several regularized MSVMs and extends the alternating direction method of multiplier (ADMM) to th…
Inexact alternating direction multiplier methods (ADMMs) are developed for solving general separable convex optimization problems with a linear constraint and with an objective that is the sum of smooth and nonsmooth terms. The approach involves linearized subproblems, a back substitution step, and either gradient or accelerated gradient techniques. Global convergence is established. The methods are particularly useful when the ADMM subproblems do not have closed form solution or when the solution of the subproblems is expensive. Numerical experiments based on image reconstruction problems sho…
In this paper, we consider a proximal linearized alternating direction method of multipliers (PL-ADMM) for solving linearly constrained nonconvex and possibly nonsmooth optimization problems. The algorithm is generalized by using variable metric proximal terms in the primal updates and an over-relaxation stepsize in the multiplier update. We prove that the sequence generated by this method is bounded and its limit points are critical points. Under the powerful Kurdyka-{Łojasiewicz} properties we prove that the sequence has a finite length thus converges, and we drive its convergence rates.
