In this paper we propose and analyze two dual methods based on inexact gradient information and averaging that generate approximate primal solutions for smooth convex optimization problems. The complicating constraints are moved into the cost using the Lagrange multipliers. The dual problem is solved by inexact first order methods based on approximate gradients and we prove sublinear rate of convergence for these methods. In particular, we provide, for the first time, estimates on the primal feasibility violation and primal and dual suboptimality of the generated approximate primal and dual solutions. Moreover, we solve approximately the inner problems with a parallel coordinate descent algorithm and we show that it has linear convergence rate. In our analysis we rely on the Lipschitz property of the dual function and inexact dual gradients. Further, we apply these methods to distributed model predictive control for network systems. By tightening the complicating constraints we are also able to ensure the primal feasibility of the approximate solutions generated by the proposed algorithms. We obtain a distributed control strategy that has the following features: state and input constraints are satisfied, stability of the plant is guaranteed, whilst the number of iterations for the suboptimal solution can be precisely determined.
In this paper, we propose the inexact alternating minimization algorithm (inexact AMA), which allows inexact iterations in the algorithm, and its accelerated variant, called the inexact fast alternating minimization algorithm (inexact FAMA). We show that inexact AMA and inexact FAMA are equivalent to the inexact proximal-gradient method and its accelerated variant applied to the dual problem. Based on this equivalence, we derive complexity upper-bounds on the number of iterations for the inexact algorithms. We apply inexact AMA and inexact FAMA to distributed optimization problems, with an emp…
We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information. Moreover, since the exact solution of the augmented Lagrangian primal problem is hard to compute in practice, we solve this problem up to some given inner accuracy. We derive relations between the inner and the outer accuracy of the primal…
This paper develops a distributed model predictive control (DMPC) strategy for a class of discrete-time linear systems with consideration of globally coupled constraints. The DMPC under study is based on the dual problem concerning all subsystems, which is solved by means of the primal-dual gradient optimization in a distributed manner using Laplacian consensus. To reduce the computational burden, the constraint tightening method is utilized to provide a capability of premature termination with guaranteeing the convergence of the DMPC optimization. The contraction theory is first adopted in th…
We present and analyze a stochastic distributed method (S-NEAR-DGD) that can tolerate inexact computation and inaccurate information exchange to alleviate the problems of costly gradient evaluations and bandwidth-limited communication in large-scale systems. Our method is based on a class of flexible, distributed first order algorithms that allow for the trade-off of computation and communication to best accommodate the application setting. We assume that all the information exchange between nodes is subject to random distortion and that only stochastic approximations of the true gradients are…
An Inexact Alternating Direction Method of Multipliers for Constrained Parabolic Optimal Distributed Control Problems
Jilin University（吉林大学）；Lanzhou University（兰州大学）；Beijing University of Post and Telecommunications（北京邮电大学）；Key Laboratory of Mathematics and Information Networks (Beijing University of Posts and Telecommunications), Ministry of Education（教育部数学与信息网络重点实验室（北京邮电大学））
In this paper we study inexact dumped Newton method implemented in a distributed environment. We start with an original DiSCO algorithm [Communication-Efficient Distributed Optimization of Self-Concordant Empirical Loss, Yuchen Zhang and Lin Xiao, 2015]. We will show that this algorithm may not scale well and propose an algorithmic modifications which will lead to less communications, better load-balancing and more efficient computation. We perform numerical experiments with an regularized empirical loss minimization instance described by a 273GB dataset.
Efficient Gradient Tracking Algorithms for Distributed Optimization Problems with Inexact Communication
School of Mathematics, China University of Mining and Technology（中国矿业大学数学学院）；School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）
Application of Optimal Inexact Second-Order Acceleration to Distributed Stochastic Optimization under Statistical Similarity
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Innopolis University（因诺波利斯大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
In this paper, we study the distributed optimization problem using approximate first-order information. We suppose the agent can repeatedly call an inexact first-order oracle of each individual objective function and exchange information with its time-varying neighbors. We revisit the distributed subgradient method in this circumstance and show its suboptimality under square summable but not summable step sizes. We also present several conditions on the inexactness of the local oracles to ensure an exact convergence of the iterative sequences towards the global optimal solution. A numerical ex…
MUSIC: Accelerated Convergence for Distributed Optimization With Inexact and Exact Methods
School of Computer Science and Technology, Hubei University of Science and Technology（湖北科技学院计算机科学与技术学院）；School of Electronic and Electrical Engineering, Wuhan Textile University（武汉纺织大学电子与电气工程学院）；The University of Manchester（曼彻斯特大学）
Dual decomposition is widely utilized in distributed optimization of multi-agent systems. In practice, the dual decomposition algorithm is desired to admit an asynchronous implementation due to imperfect communication, such as time delay and packet drop. In addition, computational errors also exist when individual agents solve their own subproblems. In this paper, we analyze the convergence of the dual decomposition algorithm in distributed optimization when both the asynchrony in communication and the inexactness in solving subproblems exist. We find that the interaction between asynchrony an…
This paper consists of a tutorial on the Augmented Lagrangian based Alternating Direction Inexact Newton method (ALADIN) and its application to distributed model predictive control (MPC). The focus is - for simplicity of presentation - on convex quadratic programming (QP) formulations of MPC. It is explained how ALADIN can be used to synthesize sparse QP solvers for large-scale linear-quadratic optimal control by combining ideas from augmented Lagrangian methods, sequential quadratic programming, as well as barrier or interior point methods. The highlight of this tutorial is a real-time ALADIN…
In this two-part work, we propose an algorithmic framework for solving non-convex problems whose objective function is the sum of a number of smooth component functions plus a convex (possibly non-smooth) or/and smooth (possibly non-convex) regularization function. The proposed algorithm incorporates ideas from several existing approaches such as alternate direction method of multipliers (ADMM), successive convex approximation (SCA), distributed and asynchronous algorithms, and inexact gradient methods. Different from a number of existing approaches, however, the proposed framework is flexible…
Optimal Design of Distributed Inexact Gradient Tracking: Exact Worst-case Convergence Rate and Explicit Parameters
State Key Laboratory of Industrial Control Technology and the College of Control Science and Engineering, Zhejiang University（浙江大学工业控制技术国家重点实验室与控制科学学院）
We consider the structured stochastic convex program requiring the minimization of $\mathbb{E}[\tilde f(x,ξ)]+\mathbb{E}[\tilde g(y,ξ)]$ subject to the constraint $Ax + By = b$. Motivated by the need for decentralized schemes and structure, we propose a stochastic inexact ADMM (SI-ADMM) framework where subproblems are solved inexactly via stochastic approximation schemes. Based on this framework, we prove the following: (i) under suitable assumptions on the associated batch-size of samples utilized at each iteration, the SI-ADMM scheme produces a sequence that converges to the unique solution …
In this paper we propose a parallel coordinate descent algorithm for solving smooth convex optimization problems with separable constraints that may arise e.g. in distributed model predictive control (MPC) for linear network systems. Our algorithm is based on block coordinate descent updates in parallel and has a very simple iteration. We prove (sub)linear rate of convergence for the new algorithm under standard assumptions for smooth convex optimization. Further, our algorithm uses local information and thus is suitable for distributed implementations. Moreover, it has low iteration complexit…
This paper addresses the problem of seeking a common fixed point for a collection of nonexpansive operators over time-varying multi-agent networks in real Hilbert spaces, where each operator is only privately and approximately known to each individual agent, and all agents need to cooperate to solve this problem by propagating their own information to their neighbors through local communications over time-varying networks. To handle this problem, inspired by the centralized inexact Krasnosel'ski\uı-Mann (IKM) iteration, we propose a distributed algorithm, called distributed inexact Krasnosel's…
This paper presents a Distributed Stochastic Model Predictive Control algorithm for networks of linear systems with multiplicative uncertainties and local chance constraints on the states and control inputs. The chance constraints are approximated via Cantelli's inequality by means of expected value and covariance. The cooperative control algorithm is based on the distributed Alternating Direction Method of Multipliers, which renders the controller fully distributedly implementable, recursively feasible and ensures point-wise convergence of the states. The aforementioned properties are guarant…
In this paper, we present a novel stochastic output-feedback MPC scheme for distributed systems with additive process and measurement noise. The chance constraints are treated with the concept of probabilistic reachable sets, which, under an unimodality assumption on the disturbance distributions are guaranteed to be satisfied in closed-loop. By conditioning the initial state of the optimization problem on feasibility, the fundamental property of recursive feasibility is ensured. Closed-loop chance constraint satisfaction, recursive feasibility and convergence to an asymptotic average cost bou…
In this paper, we propose new proximal Newton-type methods for convex optimization problems in composite form. The applications include model predictive control (MPC) and embedded MPC. Our new methods are computationally attractive since they do not require evaluating the Hessian at each iteration while keeping fast convergence rate. More specifically, we prove the global convergence is guaranteed and the superlinear convergence is achieved in the vicinity of an optimal solution. We also develop several practical variants by incorporating quasi-Newton and inexact subproblem solving schemes and…
Time-distributed Optimization (TDO) is an approach for reducing the computational burden of Model Predictive Control (MPC). When using TDO, optimization iterations are distributed over time by maintaining a running solution estimate and updating it at each sampling instant. In this paper, TDO applied to input constrained linear MPC is studied in detail, and analytic expressions for the system gains and a bound on the number of optimization iterations per sampling instant required to guarantee closed-loop stability is derived. Further, it is shown that the closed-loop stability of TDO-based MPC…
We propose a parallel adaptive constraint-tightening approach to solve a linear model predictive control problem for discrete-time systems, based on inexact numerical optimization algorithms and operator splitting methods. The underlying algorithm first splits the original problem in as many independent subproblems as the length of the prediction horizon. Then, our algorithm computes a solution for these subproblems in parallel by exploiting auxiliary tightened subproblems in order to certify the control law in terms of suboptimality and recursive feasibility, along with closed-loop stability …
In dual decomposition, the dual to an optimization problem with a specific structure is solved in distributed fashion using (sub)gradient and recently also fast gradient methods. The traditional dual decomposition suffers from two main short-comings. The first is that the convergence is often slow, although fast gradient methods have significantly improved the situation. The second is that computation of the optimal step-size requires centralized computations, which hinders a fully distributed implementation of the algorithm. In this paper, the first issue is addressed by providing a tighter c…
We consider distributed stochastic optimization problems that are solved with master/workers computation architecture. Statistical arguments allow to exploit statistical similarity and approximate this problem by a finite-sum problem, for which we propose an inexact accelerated cubic-regularized Newton's method that achieves lower communication complexity bound for this setting and improves upon existing upper bound. We further exploit this algorithm to obtain convergence rate bounds for the original stochastic optimization problem and compare our bounds with the existing bounds in several reg…
Statistical preconditioning enables fast methods for distributed large-scale empirical risk minimization problems. In this approach, multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smaller-scale optimization problem. The recently proposed Statistically Preconditioned Accelerated Gradient (SPAG) method has complexity bounds superior to other such algorithms but requires an exact solution for computationally intensive auxiliary optimization problems at every iteration. In this paper, we …
Recent studies have shown that multi-step optimization based on Model Predictive Control (MPC) can effectively coordinate the increasing number of distributed renewable energy and storage resources in the power system. However, the computation complexity of MPC is usually high which limits its use in practical implementation. To improve the efficiency of MPC, in this paper, we apply a distributed optimization method to MPC. The approach consists of a partitioning technique based on spectral clustering that determines the best system partition and an improved Optimality Condition Decomposition …
This paper presents a distributed optimization algorithm tailored to solve optimization problems arising in smart grids. In detail, we propose a variant of the Augmented Lagrangian based Alternating Direction Inexact Newton (ALADIN) method, which comes along with global convergence guarantees for the considered class of linear-quadratic optimization problems. We establish local quadratic convergence of the proposed scheme and elaborate its advantages compared to the Alternating Direction Method of Multipliers (ADMM). In particular, we show that, at the cost of more communication, ALADIN requir…
Yerevan State University（埃里温国立大学）；Center For Scientific Innovation and Education（科学与教育创新中心）；Institute of Mathematics of National Academy of Sciences of Republic of Armenia（亚美尼亚共和国国家科学院数学研究所）；Department of Mechanical Science & Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校机械科学与工程系）
The increasing presence of large-scale distributed systems highlights the need for scalable control strategies where only local communication is required. Moreover, in safety-critical systems it is imperative that such control strategies handle constraints in the presence of disturbances. In response to this need, we present the Distributed and Localized Model Predictive Control (DLMPC) algorithm for large-scale linear systems. DLMPC is a distributed closed-loop model predictive control (MPC) scheme wherein only local state and model information needs to be exchanged between subsystems for the…
Recently, distributed dual averaging has received increasing attention due to its superiority in handling constraints and dynamic networks in multiagent optimization. However, all distributed dual averaging methods reported so far considered nonsmooth problems and have a convergence rate of $O(\frac{1}{\sqrt{t}})$. To achieve an improved convergence guarantee for smooth problems, this work proposes a second-order consensus scheme that assists each agent to locally track the global dual variable more accurately. This new scheme in conjunction with smoothness of the objective ensures that the ac…
Virginia Tech（弗吉尼亚理工大学）；California Institute of Technology（加州理工学院）
This paper presents a layered control approach for real-time trajectory planning and control of robust cooperative locomotion by two holonomically constrained quadrupedal robots. A novel interconnected network of reduced-order models, based on the single rigid body (SRB) dynamics, is developed for trajectory planning purposes. At the higher level of the control architecture, two different model predictive control (MPC) algorithms are proposed to address the optimal control problem of the interconnected SRB dynamics: centralized and distributed MPCs. The distributed MPC assumes two local quadra…
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…
Controlling Heating, Ventilation and Air Conditioning (HVAC) system to maintain occupant's indoor thermal comfort is important to energy-efficient buildings and the development of smart cities. In this paper, we formulate a model predictive controller (MPC) to make optimal control strategies to HVAC in order to maintain occupant's comfort by predicted mean vote index, and then present a whole system with an estimator to indoor climate and apartment's geometric information based on only thermostats. In order to have accurate spatial resolution and make the HVAC system focus on only a zoned area…
The increasing application of voltage source converter (VSC) high voltage direct current (VSC-HVDC) technology in power grids has raised the importance of incorporating DC grids and converters into the existing transmission network. This poses significant challenges in dealing with the resulting optimal power flow (OPF) problem. In this paper, a recently proposed nonconvex distributed optimization algorithm -- Augmented Lagrangian based Alternating Direction Inexact Newton method (ALADIN), is tailored to solve the nonconvex AC/DC OPF problem for emerging voltage source converter (VSC) based mu…
In this paper we consider a network with $P$ nodes, where each node has exclusive access to a local cost function. Our contribution is a communication-efficient distributed algorithm that finds a vector $x^\star$ minimizing the sum of all the functions. We make the additional assumption that the functions have intersecting local domains, i.e., each function depends only on some components of the variable. Consequently, each node is interested in knowing only some components of $x^\star$, not the entire vector. This allows for improvement in communication-efficiency. We apply our algorithm to m…
Anytime Plug-and-Play Control with Contract-Based Distributed MPC
Institute for Dynamic Systems and Control, ETH Zurich（动态系统与控制研究所，苏黎世联邦理工学院）；Institute of Mechanical Engineering, École Polytechnique Fédérale de Lausanne（机械工程研究所，洛桑联邦理工学院）
