In this paper we propose an efficient distributed algorithm for solving loosely coupled convex optimization problems. The algorithm is based on a primal-dual interior-point method in which we use the alternating direction method of multipliers (ADMM) to compute the primal-dual directions at each iteration of the method. This enables us to join the exceptional convergence properties of primal-dual interior-point methods with the remarkable parallelizability of ADMM. The resulting algorithm has superior computational properties with respect to ADMM directly applied to our problem. The amount of computations that needs to be conducted by each computing agent is far less. In particular, the updates for all variables can be expressed in closed form, irrespective of the type of optimization problem. The most expensive computational burden of the algorithm occur in the updates of the primal variables and can be precomputed in each iteration of the interior-point method. We verify and compare our method to ADMM in numerical experiments.
In this paper, we propose a distributed algorithm for solving loosely coupled problems with chordal sparsity which relies on primal-dual interior-point methods. We achieve this by distributing the computations at each iteration, using message-passing. In comparison to already existing distributed algorithms for solving such problems, this algorithm requires far less number of iterations to converge to a solution with high accuracy. Furthermore, it is possible to compute an upper-bound for the number of required iterations which, unlike already existing methods, only depends on the coupling str…
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…
In a multi-agent network, we consider the problem of minimizing an objective function that is expressed as the sum of private convex and smooth functions, and a (possibly) non-differentiable convex regularizer. We propose a novel distributed second-order method based on the framework of Alternating Direction Method of Multipliers (ADMM), by invoking approximate Newton iterations to the primal update corresponding to the differentiable part. In order to achieve a distributed implementation, the total Hessian matrix is split into a diagonal component (locally computable) and an off-diagonal comp…
We consider cooperative multi-agent resource sharing problems over time-varying communication networks, where only local communications are allowed. The objective is to minimize the sum of agent-specific composite convex functions subject to a conic constraint that couples agents' decisions. We propose a distributed primal-dual algorithm DPDA-D to solve the saddle point formulation of the sharing problem on time-varying (un)directed communication networks; and we show that primal-dual iterate sequence converges to a point defined by a primal optimal solution and a consensual dual price for the…
We propose an efficient first-order method, based on the alternating direction method of multipliers (ADMM), to solve the homogeneous self-dual embedding problem for a primal-dual pair of semidefinite programs (SDPs) with chordal sparsity. Using a series of block eliminations, the per-iteration cost of our method is the same as applying a splitting method to the primal or dual alone. Moreover, our approach is more efficient than other first-order methods for generic sparse conic programs since we work with smaller semidefinite cones. In contrast to previous first-order methods that exploit cho…
In this paper we provide a generalization of the Douglas-Rachford splitting (DRS) and the primal-dual algorithm (Vu 2013, Condat 2013) for solving monotone inclusions in a real Hilbert space involving a general linear operator. The proposed method allows for primal and dual non-standard metrics and activates the linear operator separately from the monotone operators appearing in the inclusion. In the simplest case when the linear operator has full range, it reduces to classical DRS. Moreover, the weak convergence of primal-dual sequences to a Kuhn-Tucker point is guaranteed, generalizing the m…
We consider a class of distributed optimization problem where the objective function consists of a sum of strongly convex and smooth functions and a (possibly nonsmooth) convex regularizer. A multi-agent network is assumed, where each agent holds a private cost function and cooperates with its neighbors to compute the optimum of the aggregate objective. We propose a quasi-Newton Alternating Direction Method of Multipliers (ADMM) where the primal update is solved inexactly with approximated curvature information. By introducing an intermediate consensus variable, we achieve a block diagonal Hes…
For multi-block alternating direction method of multipliers(ADMM), where the objective function can be decomposed into multiple block components, we show that with block symmetric Gauss-Seidel iteration, the algorithm will converge quickly. The method will apply a block symmetric Gauss-Seidel iteration in the primal update and a linear correction that can be derived in view of Richard iteration. We also establish the linear convergence rate for linear systems.
We extend a primal-dual fixed point algorithm (PDFP) proposed in [5] to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrate how practical and fully decoupled schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also present. We demonstrate how different algorithms can be obtained by splitting the problems in di…
This paper presents the Distributed Primal Outer Approximation (DiPOA) algorithm for solving Sparse Convex Programming (SCP) problems with separable structures, efficiently, and in a decentralized manner. The DiPOA algorithm development consists of embedding the recently proposed Relaxed Hybrid Alternating Direction Method of Multipliers (RH-ADMM) algorithm into the Outer Approximation (OA) algorithm. We also propose two main improvements to control the quality and the number of cutting planes that approximate nonlinear functions. In particular, the RH-ADMM algorithm acts as a distributed nume…
Despite the vast literature on DRS and ADMM, there has been very little work analyzing their behavior under pathologies. Most analyses assume a primal solution exists, a dual solution exists, and strong duality holds. When these assumptions are not met, i.e., under pathologies, the theory often breaks down and the empirical performance may degrade significantly. In this paper, we establish that DRS only requires strong duality to work, in the sense that asymptotically iterates are approximately feasible and approximately optimal.
We consider the convex-concave saddle point problem $\min_{\mathbf{x}}\max_{\mathbf{y}}Φ(\mathbf{x},\mathbf{y})$, where the decision variables $\mathbf{x}$ and/or $\mathbf{y}$ subject to a multi-block structure and affine coupling constraints, and $Φ(\mathbf{x},\mathbf{y})$ possesses certain separable structure. Although the minimization counterpart of such problem has been widely studied under the topics of ADMM, this minimax problem is rarely investigated. In this paper, a convenient notion of $ε$-saddle point is proposed, under which the convergence rate of several proposed algorithms are a…
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.
We develop two new variants of alternating direction methods of multipliers (ADMM) and two parallel primal-dual decomposition algorithms to solve a wide range class of constrained convex optimization problems. Our approach relies on a novel combination of the augmented Lagrangian framework, partial alternating/linearization scheme, Nesterov's acceleration technique, and adaptive strategy. The proposed algorithms have the following new features compared to existing ADMM variants. First, they have a Nesterov's acceleration step on the primal variables instead of the dual ones as in several ADMM …
We propose in this paper a unifying scheme for several algorithms from the literature dedicated to the solving of monotone inclusion problems involving compositions with linear continuous operators in infinite dimensional Hilbert spaces. We show that a number of primal-dual algorithms for monotone inclusions and also the classical ADMM numerical scheme for convex optimization problems, along with some of its variants, can be embedded in this unifying scheme. While in the first part of the paper convergence results for the iterates are reported, the second part is devoted to the derivation of c…
We employ chordal decomposition to reformulate a large and sparse semidefinite program (SDP), either in primal or dual standard form, into an equivalent SDP with smaller positive semidefinite (PSD) constraints. In contrast to previous approaches, the decomposed SDP is suitable for the application of first-order operator-splitting methods, enabling the development of efficient and scalable algorithms. In particular, we apply the alternating direction method of multipliers (ADMM) to solve decomposed primal- and dual-standard-form SDPs. Each iteration of such ADMM algorithms requires a projection…
The Potts model has many applications. It is equivalent to some min-cut and max-flow models. Primal-dual algorithms have been used to solve these problems. Due to the special structure of the models, convergence proof is still a difficult problem. In this work, we developed two novel, preconditioned, and over-relaxed alternating direction methods of multipliers (ADMM) with convergence guarantee for these models. Using the proposed preconditioners or block preconditioners, we get accelerations with the over-relaxation variants of preconditioned ADMM. The preconditioned and over-relaxed Douglas-…
Elliptic PDE-constrained optimal control problems with $L^1$-control cost ($L^1$-EOCP) are considered. To solve $L^1$-EOCP, the primal-dual active set (PDAS) method, which is a special semismooth Newton (SSN) method, used to be a priority. However, in general solving Newton equations is expensive. Motivated by the success of alternating direction method of multipliers (ADMM), we consider extending the ADMM to $L^1$-EOCP. To discretize $L^1$-EOCP, the piecewise linear finite element (FE) is considered. However, different from the finite dimensional $l^1$-norm, the discretized $L^1$-norm does no…
In this work, we consider the distributed optimization problem in which each node has its own convex cost function and can communicate directly only with its neighbors, as determined by a directed communication topology (directed graph or digraph). First, we reformulate the optimization problem so that Alternating Direction Method of Multipliers (ADMM) can be utilized. Then, we propose an algorithm, herein called Distributed Alternating Direction Method of Multipliers using Finite-Time Exact Ratio Consensus (D-ADMM-FTERC), to solve the multi-node convex optimization problem, in which every nod…
We consider constraint-coupled optimization problems in which agents of a network aim to cooperatively minimize the sum of local objective functions subject to individual constraints and a common linear coupling constraint. We propose a novel optimization algorithm that embeds a dynamic average consensus protocol in the parallel Alternating Direction Method of Multipliers (ADMM) to design a fully distributed scheme for the considered set-up. The dynamic average mechanism allows agents to track the time-varying coupling constraint violation (at the current solution estimates). The tracked versi…
Although originally designed and analyzed for convex problems, the alternating direction method of multipliers (ADMM) and its close relatives, Douglas-Rachford splitting (DRS) and Peaceman-Rachford splitting (PRS), have been observed to perform remarkably well when applied to certain classes of structured nonconvex optimization problems. However, partial global convergence results in the nonconvex setting have only recently emerged. In this paper we show how the Douglas-Rachford envelope (DRE), introduced in 2014, can be employed to unify and considerably simplify the theory for devising globa…
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；College of Humanities and Sciences（人文与科学学院）；Northeast Normal University（东北师范大学）；University of Minnesota, Twin Cities（明尼苏达大学双城分校）
Alternating Direction Method of Multipliers (ADMM) has become a widely used optimization method for convex problems, particularly in the context of data mining in which large optimization problems are often encountered. ADMM has several desirable properties, including the ability to decompose large problems into smaller tractable sub-problems and ease of parallelization, that are essential in these scenarios. The most common form of ADMM is the two-block, in which two sets of primal variables are updated alternatingly. Recent years have seen advances in multi-block ADMM, which update more than…
We study preconditioned proximal point methods for a class of saddle point problems, where the preconditioner decouples the overall proximal point method into an alternating primal--dual method. This is akin to the Chambolle--Pock method or the ADMM. In our work, we replace the squared distance in the dual step by a barrier function on a symmetric cone, while using a standard (Euclidean) proximal step for the primal variable. We show that under non-degeneracy and simple linear constraints, such a hybrid primal--dual algorithm can achieve linear convergence on originally strongly convex problem…
We introduce Newton-ADMM, a method for fast conic optimization. The basic idea is to view the residuals of consecutive iterates generated by the alternating direction method of multipliers (ADMM) as a set of fixed point equations, and then use a nonsmooth Newton method to find a solution; we apply the basic idea to the Splitting Cone Solver (SCS), a state-of-the-art method for solving generic conic optimization problems. We demonstrate theoretically, by extending the theory of semismooth operators, that Newton-ADMM converges rapidly (i.e., quadratically) to a solution; empirically, Newton-ADMM…
In the present paper we propose a novel convergence analysis of the Alternating Direction Methods of Multipliers (ADMM), based on its equivalence with the overrelaxed Primal-Dual Hybrid Gradient (oPDHG) algorithm. We consider the smooth case, which correspond to the cas where the objective function can be decomposed into one differentiable with Lipschitz continuous gradient part and one strongly convex part. An accelerated variant of the ADMM is also proposed, which is shown to converge linearly with same rate as the oPDHG.
Primal-dual hybrid gradient method (PDHG, a.k.a. Chambolle and Pock method) is a well-studied algorithm for minimax optimization problems with a bilinear interaction term. Recently, PDHG is used as the base algorithm for a new LP solver PDLP that aims to solve large LP instances by taking advantage of modern computing resources, such as GPU and distributed system. Most of the previous convergence results of PDHG are either on duality gap or on distance to the optimal solution set, which are usually hard to compute during the solving process. In this paper, we propose a new progress metric for …
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.
This paper focuses on the decentralized optimization problem, where agents in a network cooperate to minimize the sum of their local objective functions by information exchange and local computation. Based on the alternating direction method of multipliers (ADMM), we propose CC-DQM, a communication-efficient decentralized second-order optimization algorithm that combines compressed communication with event-triggered communication. Specifically, agents are allowed to transmit the compressed message only when the current primal variables have changed greatly compared to its last estimate. Moreov…
On the Local Linear Convergence of Operator Splitting Methods for Conic Programming
University of California San Diego（加州大学圣地亚哥分校）；MIT, Sloan School of Management（麻省理工学院斯隆管理学院）；University of Chicago, Department of Statistics（芝加哥大学统计学系）
Elliptic optimal control problems with pointwise box constraints on the control (EOCP) are considered. To solve EOCP, the primal-dual active set (PDAS) method, which is a special semismooth Newton (SSN) method, used to be a priority in consideration of their locally superlinear convergence. However, in general solving the Newton equations is expensive, especially when the discretization is in a fine level. Motivated by the success of applying alternating direction method of multipliers (ADMM) for solving large scale convex minimization problem in finite dimension, it is reasonable to extend th…
Many problems in control theory can be formulated as semidefinite programs (SDPs). For large-scale SDPs, it is important to exploit the inherent sparsity to improve the scalability. This paper develops efficient first-order methods to solve SDPs with chordal sparsity based on the alternating direction method of multipliers (ADMM). We show that chordal decomposition can be applied to either the primal or the dual standard form of a sparse SDP, resulting in scaled versions of ADMM algorithms with the same computational cost. Each iteration of our algorithms consists of a projection on the produc…
In this paper, we show that for a class of linearly constrained convex composite optimization problems, an (inexact) symmetric Gauss-Seidel based majorized multi-block proximal alternating direction method of multipliers (ADMM) is equivalent to an {\em inexact} proximal augmented Lagrangian method (ALM). This equivalence not only provides new perspectives for understanding some ADMM-type algorithms but also supplies meaningful guidelines on implementing them to achieve better computational efficiency. Even for the two-block case, a by-product of this equivalence is the convergence of the whole…
This paper is devoted to the design of an efficient and convergent {semi-proximal} alternating direction method of multipliers (ADMM) for finding a solution of low to medium accuracy to convex quadratic conic programming and related problems. For this class of problems, the convergent two block semi-proximal ADMM can be employed to solve their primal form in a straightforward way. However, it is known that it is more efficient to apply the directly extended multi-block semi-proximal ADMM, though its convergence is not guaranteed, to the dual form of these problems. Naturally, one may ask the f…
The alternating direction method of multipliers (ADMM) has been recently recognized as a promising optimizer for large-scale machine learning models. However, there are very few results studying ADMM from the aspect of communication costs, especially jointly with privacy preservation, which are critical for distributed learning. We investigate the communication efficiency and privacy-preservation of ADMM by solving the consensus optimization problem over decentralized networks. Since walk algorithms can reduce communication load, we first propose incremental ADMM (I-ADMM) based on the walk alg…
In this paper, we study the local linear convergence properties of a versatile class of Primal-Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework we first show that (i) the sequences generated by Primal-Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a loc…
We propose an early termination technique for mixed integer conic programming for use within branch-and-bound based solvers. Our approach generalizes previous early termination results for ADMM-based solvers to a broader class of primal-dual algorithms, including both operator splitting methods and interior point methods. The complexity for checking early termination is $O(n)$ for each termination check assuming a bounded problem domain. We show that this domain restriction can be relaxed for problems whose data satisfies a simple rank condition, in which case each check requires an $O(n^2)$ s…
In this work, we introduce ADAPD, $\textbf{A}$ $\textbf{D}$ecentr$\textbf{A}$lized $\textbf{P}$rimal-$\textbf{D}$ual algorithmic framework for solving non-convex and smooth consensus optimization problems over a network of distributed agents. The proposed framework relies on a novel problem formulation that elicits ADMM-type updates, where each agent first inexactly solves a local strongly convex subproblem with any method of its choice and then performs a neighbor communication to update a set of dual variables. We present two variants that allow for a single gradient step for the primal upda…
Alternating direction methods of multipliers (ADMMs) are popular approaches to handle large scale semidefinite programs that gained attention during the past decade. In this paper, we focus on solving doubly nonnegative programs (DNN), which are semidefinite programs where the elements of the matrix variable are constrained to be nonnegative. Starting from two algorithms already proposed in the literature on conic programming, we introduce two new ADMMs by employing a factorization of the dual variable. It is well known that first order methods are not suitable to compute high precision optima…
In this paper, we propose two exact distributed algorithms to solve mixed integer linear programming (MILP) problems with multiple agents where data privacy is important for the agents. A key challenge is that, because of the non-convex nature of MILPs, classical distributed and decentralized optimization approaches cannot be applied directly to find their optimal solutions. The proposed exact algorithms are based on adding primal cuts and restricting the Lagrangian relaxation of the original MILP problem. We show finite convergence of these algorithms for MILPs with only binary and continuous…
