An inertial alternating direction method of multipliers for solving a two-block separable convex minimization problem
Yang Yang, Yuchao Tang
备注：
22 pages
发表信息：
An Inertial Alternating Direction Method of Multipliers for Solving a Two-Block Separable Convex Minimization Problem[J].Journal of Mathematical Research with Applications,2021,41(2):204-220
The alternating direction method of multipliers (ADMM) is a widely used method for solving many convex minimization models arising in signal and image processing. In this paper, we propose an inertial ADMM for solving a two-block separable convex minimization problem with linear equality constraints. This algorithm is obtained by making use of the inertial Douglas-Rachford splitting algorithm to the corresponding dual of the primal problem. We study the convergence analysis of the proposed algorithm in infinite-dimensional Hilbert spaces. Furthermore, we apply the proposed algorithm on the robust principal component pursuit problem and also compare it with other state-of-the-art algorithms. Numerical results demonstrate the advantage of the proposed algorithm.
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 propose a second-order continuous primal-dual dynamical system with time-dependent positive damping terms for a separable convex optimization problem with linear equality constraints. By the Lyapunov function approach, we investigate asymptotic properties of the proposed dynamical system as the time $t\to+\infty$. The convergence rates are derived for different choices of the damping coefficients. We also show that the obtained results are robust under external perturbations.
To solve the separable convex optimization problem with linear constraints, Eckstein and Bertsekas introduced the generalized alternating direction method of multipliers (in short, GADMM), which is an efficient and simple acceleration scheme of the aternating direction method of multipliers. Recently, \textbf{Fang et. al} proposed the linearized version of generalized alternating direction method of multipliers (in short, L-GADMM), where one of its subproblems is approximated by a linearization strategy, and proved its worst-case $\mathcal{O}(1/t)$ convergence rate measured by the iteration co…
This paper proposes a two-point inertial proximal point algorithm to find zero of maximal monotone operators in Hilbert spaces. We obtain weak convergence results and non-asymptotic $O(1/n)$ convergence rate of our proposed algorithm in non-ergodic sense. Applications of our results to various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers are given. Numerical results are given to demonstrate the accelerating behaviors of our method over other related methods in the literature.
The alternating direction method of multipliers (ADMM) is one of the most widely used first-order optimisation methods in the literature owing to its simplicity, flexibility and efficiency. Over the years, numerous efforts are made to improve the performance of the method, such as the inertial technique. By studying the geometric properties of ADMM, we discuss the limitations of current inertial accelerated ADMM and then present and analyze an adaptive acceleration scheme for the method. Numerical experiments on problems arising from image processing, statistics and machine learning demonstrat…
Convergence analysis of accelerated algorithms via a mixed-order dynamical system for separable nonsmooth convex optimization
Chongqing Key Laboratory of Statistical Intelligent Computing and Monitoring, School of Mathematics and Statistics, Chongqing Technology and Business University（重庆工商大学数学与统计学院，统计智能计算与监测重庆市重点实验室）
In this paper, we first propose a general inertial proximal point method for the mixed variational inequality (VI) problem. Based on our knowledge, without stronger assumptions, convergence rate result is not known in the literature for inertial type proximal point methods. Under certain conditions, we are able to establish the global convergence and a $o(1/k)$ convergence rate result (under certain measure) of the proposed general inertial proximal point method. We then show that the linearized alternating direction method of multipliers (ADMM) for separable convex optimization with linear co…
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…
We consider a multi-block separable convex optimization problem with the linear constraints, where the objective function is the sum of m individual convex functions without overlapping variables. The linearized version of the generalized alternating direction method of multipliers (L-GADMM) is particularly efficient for the two-block separable convex programming problem and its convergence was proved when two blocks of variables are alternatively updated. However,the convergence and some practical applications of the extension (m>=3) of the L-GADMM is still in its infancy. In this paper, we e…
This paper derives new inexact variants of the Douglas-Rachford splitting method for maximal monotone operators and the alternating direction method of multipliers (ADMM) for convex optimization. The analysis is based on a new inexact version of the proximal point algorithm that includes both an inertial step and overrelaxation. We apply our new inexact ADMM method to LASSO and logistic regression problems and obtain somewhat better computational performance than earlier inexact ADMM methods.
The alternating direction method of multipliers (ADMM) is a most widely used optimization scheme for solving linearly constrained separable convex optimization problems. The convergence of the ADMM can be guaranteed when the dual step length is less than the golden ratio, while plenty of numerical evidence suggests that even larger dual step length often accelerates the convergence. It has also been proved that the dual step length can be enlarged to less than 2 in some special cases, namely, one of the separable functions in the objective function is linear, or both are quadratic plus some ad…
In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case O(1/t) convergence rate, wheret denotes the iteration number. By properly choosing the algorithm parameters, numerical experiments on solving a sparse optimization problem arising from statistical learning show that our P-PPA could perform significantly better than other state-of-the-art methods, such as the alternating direction method of m…
The alternating direction method of multipliers (ADMM) is a popular method for solving convex separable minimization problems with linear equality constraints. The generalization of the two-block ADMM to the three-block ADMM is not trivial since the three-block ADMM is not convergence in general. Many variants of three-block ADMM have been developed with guarantee convergence. Besides the ADMM, the alternating minimization algorithm (AMA) is also an important algorithm for solving the convex separable minimization problem with linear equality constraints. The AMA is first proposed by Tseng, an…
In the past decade, we had developed a series of splitting contraction algorithms for separable convex optimization problems, at the root of the alternating direction method of multipliers. Convergence of these algorithms was studied under specific model-tailored conditions, while these conditions can be conceptually abstracted as two generic conditions when these algorithms are all unified as a prediction-correction framework. In this paper, in turn, we showcase a constructive way for specifying the generic convergence-guaranteeing conditions, via which new splitting contraction algorithms ca…
The Alternating Direction Method of Multipliers (ADMM) has been proved to be effective for solving separable convex optimization subject to linear constraints. In this paper, we propose a Generalized Symmetric ADMM (GS-ADMM), which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming. This GS-ADMM partitions the data into two group variables so that one group consists of $p$ block variables while the other has $q$ block variables, where $p \ge 1$ and $q \ge 1$ are two integers. The two grouped variables are updated in a {\it Gauss…
We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the gl…
We consider the problem of minimizing block-separable convex functions subject to linear constraints. While the Alternating Direction Method of Multipliers (ADMM) for two-block linear constraints has been intensively studied both theoretically and empirically, in spite of some preliminary work, effective generalizations of ADMM to multiple blocks is still unclear. In this paper, we propose a randomized block coordinate method named Parallel Direction Method of Multipliers (PDMM) to solve the optimization problems with multi-block linear constraints. PDMM randomly updates some primal and dual b…
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…
The linearly constrained convex composite programming problems whose objective function contains two blocks with each block being the form of nonsmooth+smooth arises frequently in multiple fields of applications. If both of the smooth terms are quadratic, this problem can be solved efficiently by using the symmetric Gaussian-Seidel (sGS) technique based proximal alternating direction method of multipliers (ADMM). However, in the non-quadratic case, the sGS technique can not be used any more, which leads to the separable structure of nonsmooth+smooth had to be ignored. In this paper, we present…
We study stochastic convex optimization subjected to linear equality constraints. Traditional Stochastic Alternating Direction Method of Multipliers and its Nesterov's acceleration scheme can only achieve ergodic O(1/\sqrt{K}) convergence rates, where K is the number of iteration. By introducing Variance Reduction (VR) techniques, the convergence rates improve to ergodic O(1/K). In this paper, we propose a new stochastic ADMM which elaborately integrates Nesterov's extrapolation and VR techniques. We prove that our algorithm can achieve a non-ergodic O(1/K) convergence rate which is optimal fo…
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Multi-block separable convex problems recently received considerable attention. This class of optimization problems minimizes a separable convex objective function with linear constraints. The algorithmic challenges come from the fact that the classic alternating direction method of multipliers (ADMM) for the problem is not necessarily convergent. However, it is observed that ADMM outperforms numerically many of its variants with guaranteed theoretical convergence. The goal of this paper is to develop convergent and computationally efficient algorithms for solving multi-block separable convex …
Mean-variance portfolio optimization problems often involve separable nonconvex terms, including penalties on capital gains, integer share constraints, and minimum position and trade sizes. We propose a heuristic algorithm for such problems based on the alternating direction method of multipliers (ADMM). This method allows for solve times in tens to hundreds of milliseconds with around 1000 securities and 100 risk factors. We also obtain a bound on the achievable performance. Our heuristic and bound are both derived from similar results for other optimization problems with a separable objectiv…
The alternating direction method of multipliers (ADMM) proposed by Glowinski and Marrocco is a benchmark algorithm for two-block separable convex optimization problems with linear equality constraints. It has been modified, specified, and generalized from various perspectives to tackle more concrete or complicated application problems. Despite its versatility and phenomenal popularity, it remains unknown whether or not the ADMM can be extended to separable convex optimization problems with linear inequality constraints. In this paper, we lay down the foundation of how to extend the ADMM to two…
We propose a distributed algorithm, named Distributed Alternating Direction Method of Multipliers (D-ADMM), for solving separable optimization problems in networks of interconnected nodes or agents. In a separable optimization problem there is a private cost function and a private constraint set at each node. The goal is to minimize the sum of all the cost functions, constraining the solution to be in the intersection of all the constraint sets. D-ADMM is proven to converge when the network is bipartite or when all the functions are strongly convex, although in practice, convergence is observe…
An inexact accelerated stochastic Alternating Direction Method of Multipliers (AS-ADMM) scheme is developed for solving structured separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and a smooth function which is an average of many component convex functions. Problems having this structure often arise in machine learning and data mining applications. AS-ADMM combines the ideas of both ADMM and the stochastic gradient methods using variance reduction techniques. One of the ADMM subproblems employs a lineariza…
In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of the iteration number $k$, we establish global $\mathcal{O} (1/ \sqrt{k})$ pointwise and $\mathcal{O} (1/ {k})$ ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case.…
Consider the problem of minimizing the sum of a smooth convex function and a separable nonsmooth convex function subject to linear coupling constraints. Problems of this form arise in many contemporary applications including signal processing, wireless networking and smart grid provisioning. Motivated by the huge size of these applications, we propose a new class of first order primal-dual algorithms called the block successive upper-bound minimization method of multipliers (BSUM-M) to solve this family of problems. The BSUM-M updates the primal variable blocks successively by minimizing local…
This paper is concerned with two-block separable convex minimization problems with linear constraints, for which it is either impossible or too expensive to obtain the exact solutions of the subproblems involved in the proximal ADMM (alternating direction method of multipliers). Such structured convex minimization problems often arise from the two-block regroup settlement of three or four-block separable convex optimization problems with linear constraints, or from the constrained total-variation superresolution image reconstruction problems in image processing. For them, we propose an inexact…
The Alternating Direction Method of Multipliers (ADMM) has gained a lot of attention for solving large-scale and objective-separable constrained optimization. However, the two-block variable structure of the ADMM still limits the practical computational efficiency of the method, because one big matrix factorization is needed at least once even for linear and convex quadratic programming. This drawback may be overcome by enforcing a multi-block structure of the decision variables in the original optimization problem. Unfortunately, the multi-block ADMM, with more than two blocks, is not guarant…
The objective of this paper is to design an efficient and convergent alternating direction method of multipliers (ADMM) for finding a solution of medium accuracy to conic programming problems whose constraints consist of linear equalities, linear inequalities, a non-polyhedral cone and a polyhedral cone. For this class of problems, one may apply the directly extended ADMM to their dual, which can be written in the form of convex programming with four separable blocks in the objective function and a coupling linear equation constraint. Indeed, the directly extended ADMM, though may diverge in t…
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 multipliers (abbreviated as ADM). In this paper, we further clarify this connection and show that CPAs generate exactly the same sequence of points with the so-called linearized ADM (abbreviated as LADM) applied to either the primal problem …
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We propose BIBPA, a block inertial Bregman proximal algorithm for minimizing the sum of a block relatively smooth function (that is, relatively smooth concerning each block) and block separable nonsmooth nonconvex functions. We prove that the sequence generated by BIBPA subsequentially converges to critical points of the objective under standard assumptions, and globally converges when the objective function is additionally assumed to satisfy the Kurdyka-Łojasiewicz (KŁ) property. We also provide the convergence rate when the objective satisfies the Łojasiewicz inequality. We apply BIBPA to th…
Alternating Direction Method of Multipliers for nonlinear constrained convex problems and applications to distributed resource allocation and constrained machine learning
In this paper, a centralized two-block separable optimization is considered for which a fully parallel primal-dual discrete-time algorithm with fixed step size is derived based on monotone operator splitting method. In this algorithm, the primal variables are updated in an alternating fashion like Alternating Direction Method of Multipliers (ADMM). However, unlike existing discrete-time algorithms such as Method of Multipliers (MM), ADMM, Bi-Alternating Direction Method of Multipliers (BiADMM), and Primal-Dual Fixed Point (PDFP) algorithms, that all suffer from sequential updates, all primal a…
An alternating direction method of multipliers (ADMM) solver is described for optimal resource allocation problems with separable convex quadratic costs and constraints and linear coupling constraints. We describe a parallel implementation of the solver on a graphics processing unit (GPU) using a bespoke quartic function minimizer. An application to robust optimal energy management in hybrid electric vehicles is described, and the results of numerical simulations comparing the computation times of the parallel GPU implementation with those of an equivalent serial implementation are presented.
Recently, the alternating direction method of multipliers (ADMM) has found many efficient applications in various areas; and it has been shown that the convergence is not guaranteed when it is directly extended to the multiple-block case of separable convex minimization problems where there are $m\ge 3$ functions without coupled variables in the objective. This fact has given great impetus to investigate various conditions on both the model and the algorithm's parameter that can ensure the convergence of the direct extension of ADMM (abbreviated as "e-ADMM"). Despite some results under very st…
We present two efficient numerical methods for susceptibility artifact correction applicable in Echo Planar Imaging (EPI), an ultra fast Magnetic Resonance Imaging (MRI) technique widely used in clinical applications. Both methods address a major practical drawback of EPI, the so-called susceptibility artifacts, which consist of geometrical transformations and intensity modulations. We consider a tailored variational image registration problem that is based on a physical distortion model and aims at minimizing the distance of two oppositely distorted images subject to invertibility constraints…
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…
We present a flexible Alternating Direction Method of Multipliers (F-ADMM) algorithm for solving optimization problems involving a strongly convex objective function that is separable into $n \geq 2$ blocks, subject to (non-separable) linear equality constraints. The F-ADMM algorithm uses a Gauss-Seidel scheme to update blocks of variables, and a regularization term is added to each of the subproblems arising within F-ADMM. We prove, under common assumptions, that F-ADMM is globally convergent. We also present a special case of F-ADMM that is partially parallelizable, which makes it attractive…
We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty with an outlook for further applicatio…
