Improved Hierarchical ADMM for Nonconvex Cooperative Distributed Model Predictive Control
National University of Singapore（新加坡国立大学）；University of California, Berkeley（加州大学伯克利分校）；Temasek Laboratories, National University of Singapore（新加坡国立大学淡马锡实验室）；University of British Columbia（不列颠哥伦比亚大学）
Xiaoxue Zhang, Jun Ma, Zilong Cheng, Sunan Huang, Clarence W. de Silva, Tong Heng Lee
Distributed optimization is often widely attempted and innovated as an attractive and preferred methodology to solve large-scale problems effectively in a localized and coordinated manner. Thus, it is noteworthy that the methodology of distributed model predictive control (DMPC) has become a promising approach to achieve effective outcomes, e.g., in decision-making tasks for multi-agent systems. However, the typical deployment of such distributed MPC frameworks would lead to the involvement of nonlinear processes with a large number of nonconvex constraints. To address this important problem, the development and innovation of a hierarchical three-block alternating direction method of multipliers (ADMM) approach is presented in this work to solve this nonconvex cooperative DMPC problem in multi-agent systems. Here firstly, an additional slack variable is introduced to transform the original large-scale nonconvex optimization problem. Then, a hierarchical ADMM approach, which contains outer loop iteration by the augmented Lagrangian method (ALM) and inner loop iteration by three-block semi-proximal ADMM, is utilized to solve the resulting transformed nonconvex optimization problem. Additionally, it is analytically shown and established that the requisite desired stationary point exists for convergence in the algorithm. Finally, an approximate optimization stage with a barrier method is then applied to further significantly improve the computational efficiency, yielding the final improved hierarchical ADMM. The effectiveness of the proposed method in terms of attained performance and computational efficiency is demonstrated on a cooperative DMPC problem of decision-making process for multiple unmanned aerial vehicles (UAVs).
The Hong Kong University of Science and Technology（香港科技大学）
Developments in cooperative trajectory planning of connected autonomous vehicles (CAVs) have gathered considerable momentum and research attention. Generally, such problems present strong non-linearity and non-convexity, rendering great difficulties in finding the optimal solution. Existing methods typically suffer from low computational efficiency, and this hinders the appropriate applications in large-scale scenarios involving an increasing number of vehicles. To tackle this problem, we propose a novel decentralized iterative linear quadratic regulator (iLQR) algorithm by leveraging the dual…
The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various composite problems. The convergence of the conventional ADMM (i.e., 2-block) for convex objective functions has been justified for a long time, and its convergence for nonconvex objective functions has, however, been established very recently. The multi-block ADMM, a natural extension of ADMM, is a widely used scheme and has also been found very useful in solving various nonconvex optimization problems. It is thus expected to establish convergence theory of the mult…
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).
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…
This paper establishes convergence rate bounds for a variant of the proximal alternating direction method of multipliers (ADMM) for solving nonconvex linearly constrained optimization problems. The variant of the proximal ADMM allows the inclusion of an over-relaxation stepsize parameter belonging to the interval $(0,2)$. To the best of our knowledge, all related papers in the literature only consider the case where the over-relaxation parameter lies in the interval $(0,(1+\sqrt{5})/2)$.
University of Maryland（马里兰大学）；Instituto de Telecomunicações（电信研究所）；Instituto Superior Técnico（高等技术学院）；Universidade de Lisboa（里斯本大学）；Cornell University（康奈尔大学）
The alternating direction method of multipliers (ADMM) is a common optimization tool for solving constrained and non-differentiable problems. We provide an empirical study of the practical performance of ADMM on several nonconvex applications, including l0 regularized linear regression, l0 regularized image denoising, phase retrieval, and eigenvector computation. Our experiments suggest that ADMM performs well on a broad class of non-convex problems. Moreover, recently proposed adaptive ADMM methods, which automatically tune penalty parameters as the method runs, can improve algorithm efficien…
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…
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…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
With the large rising of complex data, the nonconvex models such as nonconvex loss function and nonconvex regularizer are widely used in machine learning and pattern recognition. In this paper, we propose a class of mini-batch stochastic ADMMs (alternating direction method of multipliers) for solving large-scale nonconvex nonsmooth problems. We prove that, given an appropriate mini-batch size, the mini-batch stochastic ADMM without variance reduction (VR) technique is convergent and reaches a convergence rate of $O(1/T)$ to obtain a stationary point of the nonconvex optimization, where $T$ den…
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.
The alternating direction multiplier method (ADMM) is widely used in computer graphics for solving optimization problems that can be nonsmooth and nonconvex. It converges quickly to an approximate solution, but can take a long time to converge to a solution of high-accuracy. Previously, Anderson acceleration has been applied to ADMM, by treating it as a fixed-point iteration for the concatenation of the dual variables and a subset of the primal variables. In this paper, we note that the equivalence between ADMM and Douglas-Rachford splitting reveals that ADMM is in fact a fixed-point iteration…
This paper proposes and analyzes a dampened proximal alternating direction method of multipliers (DP.ADMM) for solving linearly-constrained nonconvex optimization problems where the smooth part of the objective function is nonseparable. Each iteration of DP.ADMM consists of: (i) a sequence of partial proximal augmented Lagrangian (AL) updates, (ii) an under-relaxed Lagrange multiplier update, and (iii) a novel test to check whether the penalty parameter of the AL function should be updated. Under a basic Slater condition and some requirements related to the dampening factor and under-relaxatio…
In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, $ϕ(x_0,\ldots,x_p,y)$, subject to coupled linear equality constraints. Our ADMM updates each of the primal variables $x_0,\ldots,x_p,y$, followed by updating the dual variable. We separate the variable $y$ from $x_i$'s as it has a special role in our analysis. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, $\ell_q$ quasi-norm, Schatten-$q$ quasi-norm ($0<q<1$), m…
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…
Recently, heuristics based on the Douglas-Rachford splitting algorithm and the alternating direction method of multipliers (ADMM) have found empirical success in minimizing convex functions over nonconvex sets, but not much has been done to improve the theoretical understanding of them. In this paper, we investigate convergence of these heuristics. First, we characterize optimal solutions of minimization problems involving convex cost functions over nonconvex constraint sets. We show that these optimal solutions are related to the fixed point set of the underlying nonconvex Douglas-Rachford op…
Linearized alternating direction method of multipliers (ADMM) as an extension of ADMM has been widely used to solve linearly constrained problems in signal processing, machine leaning, communications, and many other fields. Despite its broad applications in nonconvex optimization, for a great number of nonconvex and nonsmooth objective functions, its theoretical convergence guarantee is still an open problem. In this paper, we propose a two-block linearized ADMM and a multi-block parallel linearized ADMM for problems with nonconvex and nonsmooth objectives. Mathematically, we present that the …
We consider the problem of rigid registration, where we wish to jointly register multiple point sets via rigid transforms. This arises in applications such as sensor network localization, multiview registration, and protein structure determination. The least-squares estimator for this problem can be reduced to a rank-constrained semidefinite program (REG-SDP). It was recently shown that by formally applying the alternating direction method of multipliers (ADMM), we can derive an iterative solver (REG-ADMM) for REG-SDP, wherein each subproblem admits a simple closed-form solution. The empirical…
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）
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 theoretica…
By enabling the nodes or agents to solve small-sized subproblems to achieve coordination, distributed algorithms are favored by many networked systems for efficient and scalable computation. While for convex problems, substantial distributed algorithms are available, the results for the more broad nonconvex counterparts are extremely lacking. This paper develops a distributed algorithm for a class of nonconvex and nonsmooth problems featured by i) a nonconvex objective formed by both separate and composite objective components regarding the decision components of interconnected agents, ii) loc…
We consider the stochastic nested composition optimization problem where the objective is a composition of two expected-value functions. We proposed the stochastic ADMM to solve this complicated objective. In order to find an $ε$ stationary point where the expected norm of the subgradient of corresponding augmented Lagrangian is smaller than $ε$, the total sample complexity of our method is $\mathcal{O}(ε^{-3})$ for the online case and $\mathcal{O} \Bigl((2N_1 + N_2) + (2N_1 + N_2)^{1/2}ε^{-2}\Bigr)$ for the finite sum case. The computational complexity is consistent with proximal version prop…
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…
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…
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…
Nonconvex Zeroth-Order Stochastic ADMM Methods with Lower Function Query Complexity
University of Pittsburgh（匹兹堡大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Duke University（杜克大学）；University of Maryland College Park（马里兰大学帕克分校）
We investigate a class of general combinatorial graph problems, including MAX-CUT and community detection, reformulated as quadratic objectives over nonconvex constraints and solved via the alternating direction method of multipliers (ADMM). We propose two reformulations: one using vector variables and a binary constraint, and the other further reformulating the Burer-Monteiro form for simpler subproblems. Despite the nonconvex constraint, we prove the ADMM iterates converge to a stationary point in both formulations, under mild assumptions. Additionally, recent work suggests that in this latt…
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.
Second-Order KKT Guarantees for Bregman ADMM in Nonconvex and Non-Lipschitz Optimization
Department of Electrical and Computer Engineering, Iowa State University（电气与计算机工程系，爱荷华州立大学）；Department of Computer Science and Engineering, Ohio State University（计算机科学与工程系，俄亥俄州立大学）；DAMO Academy, Alibaba Group US（阿里巴巴集团美国DAMO研究院）
Connected and automated vehicles (CAVs) offer huge potential to improve the performance of automated vehicles (AVs) without communication capabilities, especially in situations when the vehicles (or agents) need to be cooperative to accomplish their maneuver. Lane change maneuvers in dense traffic, e.g., are very challenging for non-connected AVs. To alleviate this problem, we propose a holistic distributed lane change control scheme for CAVs which relies on vehicle-to-vehicle communication. The originally centralized optimal control problem is embedded into a consensus-based Alternating Direc…
Heavy computational load for solving nonconvex problems for large-scale systems or systems with real-time demands at each sample step has been recognized as one of the reasons for preventing a wider application of nonlinear model predictive control (NMPC). To improve the real-time feasibility of NMPC with input nonlinearity, we devise an innovative scheme called cloud-based computational model predictive control (MPC) by using an elaborately designed parallel multi-block alternating direction method of multipliers (ADMM) algorithm. This novel parallel multi-block ADMM algorithm is tailored to …
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,…
