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Robustification of Continuous-Time ADMM against Communication Delays under Non-Strict Convexity: A Passivity-Based Approach
Abstract
In this paper, we address a class of distributed optimization problems with non-strictly convex cost functions in the presence of communication delays between an agent and a coordinator. To this end, we focus on a continuous-time optimization algorithm that mirrors the alternating direction method of multipliers. We first redesign the algorithm so that the dynamics ensures smoothness and a sub-block for primal optimization includes stable zeros. It is then revealed that the algorithm is composed of feedback interconnection of passive systems. We next robustify the algorithm against communication delays by applying the so-called scattering transformation. The smoothness of the dynamics allows one to use the invariance principle for delay systems, and accordingly, the state trajectories are shown to converge to an optimal solution even without the strict convexity assumption. Finally, the presented method is demonstrated via simulation of an environmental-monitoring problem.
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This work investigates the distributed constrained optimization problem under inter-agent communication delays from the perspective of passivity. First, we propose a continuous-time algorithm for distributed constrained optimization with general convex objective functions. The asymptotic stability under general convexity is guaranteed by the phase lead compensation. The inequality constraints are handled by adopting a projection-free generalized Lagrangian, whose primal-dual gradient dynamics preserves passivity and smoothness, enabling the application of the LaSalle’s invariance principle in the presence of delays. Then, we incorporate the scattering transformation into the proposed algorithm to enhance the robustness against unknown and heterogeneous communication delays. Finally, a numerical example of a matching problem is provided to illustrate the results.
In this paper, we revisit primal–dual dynamics for convex optimization and present a generalization of the dynamics based on the concept of passivity. We hypothesize that supplying a stable zero to one of the integrators in the dynamics allows one to eliminate the assumption of strict convexity on the cost function. Then, we prove this hypothesis based on the passivity paradigm together with the invariance principle for Carathéodory systems. Additionally, we show that the presented algorithm is also a generalization of existing augmented Lagrangian-based primal–dual dynamics and discuss the benefit of the presented generalization in terms of noise reduction and convergence speed. Finally, the presented algorithm is demonstrated through simulation of lighting control in a building.
In this paper, we want to study how natural and engineered systems could perform complex optimizations with limited computational and communication capabilities. We adopt a continuous-time dynamical system view rooted in early work on optimization and more recently in network protocol design, and merge it with the dynamic view of distributed averaging systems. We obtain a general approach, based on the control system viewpoint, that allows to analyze and design (distributed) optimization systems converging to the solution of given convex optimization problems. The control system viewpoint provides many insights and new directions of research. We apply the framework to a distributed optimal location problem and demonstrate the natural tracking and adaptation capabilities of the system to changing constraints.
Many problems of recent interest in statistics and machine learning can be posed in the framework of convex optimization. Due to the explosion in size and complexity of modern datasets, it is increasingly important to be able to solve problems with a very large number of features or training examples. As a result, both the decentralized collection or storage of these datasets as well as accompanying distributed solution methods are either necessary or at least highly desirable. In this review, we argue that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas. The method was developed in the 1970s, with roots in the 1950s, and is equivalent or closely related to many other algorithms, such as dual decomposition, the method of multipliers, Douglas–Rachford splitting, Spingarn's method of partial inverses, Dykstra's alternating projections, Bregman iterative algorithms for ℓ1 problems, proximal methods, and others. After briefly surveying the theory and history of the algorithm, we discuss applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. We also discuss general distributed optimization, extensions to the nonconvex setting, and efficient implementation, including some details on distributed MPI and Hadoop MapReduce implementations.
Network flow control regulates the traffic between sources and links based on congestion, and plays a critical role in ensuring satisfactory performance. In recent studies, global stability has been shown for several flow control schemes. By using a passivity approach, this paper presents a unifying framework which encompasses these stability results as special cases. In addition, the new approach significantly expands the current classes of stable flow controllers by augmenting the source and link update laws with passive dynamic systems. This generality offers the possibility of optimizing the controllers, for example, to improve robustness in stability and performance with respect to time delays, unmodeled flows, and capacity variation.
Primal–dual gradient dynamics that find saddle points of a Lagrangian have been widely employed for handling constrained optimization problems. Building on existing methods, we extend the augmented primal–dual gradient dynamics (Aug-PDGD) to incorporate general convex and nonlinear inequality constraints, and we establish its semi-global exponential stability when the objective function is strongly convex. We also provide an example of a strongly convex quadratic program of which the Aug-PDGD fails to achieve global exponential stability. Numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the KKT point.
This paper addresses a convex optimization problem that suffers from communication delays between a central collector and subsystems. First, we introduce continuous-time optimization dynamics that mirror the alternating direction method of multipliers (ADMM) architecture. Furthermore, we reveal that the proposed optimization dynamics have an architecture similar to that of bilateral teleoperation systems. By virtue of this passivity-based perspective, we apply a scattering transformation, which compensates for communication delays. Finally, we demonstrate the effectiveness of the proposed dynamics through numerical simulations of a grasp force optimization problem.
In distributed optimization of multi-agent systems, agents cooperate to minimize a global function which is a sum of local objective functions. Motivated by applications including power systems, sensor networks, smart buildings, and smart manufacturing, various distributed optimization algorithms have been developed. In these algorithms, each agent performs local computation based on its own information and information received from its neighboring agents through the underlying communication network, so that the optimization problem can be solved in a distributed manner. This survey paper aims to offer a detailed overview of existing distributed optimization algorithms and their applications in power systems. More specifically, we first review discrete-time and continuous-time distributed optimization algorithms for undirected graphs. We then discuss how to extend these algorithms in various directions to handle more realistic scenarios. Finally, we focus on the application of distributed optimization in the optimal coordination of distributed energy resources.
This letter is devoted to the concept of
instant
model predictive control (iMPC) for linear systems. An optimization problem is formulated to express the finite-time constrained optimal regulation control, like conventional Model predictive control (MPC). Then, iMPC determines the control action based on the optimization process rather than the optimizer, unlike MPC. The iMPC concept is realized by a continuous-time dynamic algorithm of solving the optimization; the primal-dual gradient algorithm is directly implemented as a dynamic controller. On the basis of the dissipativity evaluation of the algorithm, the stability of the control system is analyzed. Finally, a numerical experiment is performed in order to demonstrate that iMPC emulates MPC and to show its less computational burden.
This paper deals with passivity-based stability analysis of electricity market trading with dynamic pricing. In the deregulated electricity market, power consumers and generators will participate in market trading as market players. For such a new kind of the market trading system, the dynamic pricing procedure has to be taken into account of the intermittent participation of power consumers and generators. Then, this paper discusses the stability of the electricity market trading system including power flow using passivity analysis. This paper also shows that the optimal power demand, supply and electricity prices are asymptotically stable and these values are derived in a distributed manner through market trading.
Highlighting the control of networked robotic systems, this book synthesizes a unified passivity-based approach to an emerging cross-disciplinary subject. Thanks to this unified approach, readers can access various state-of-the-art research fields by studying only the background foundations associated with passivity. In addition to the theoretical results and techniques,the authors provide experimental case studies on testbeds of robotic systems including networked haptic devices, visual robotic systems, robotic network systems and visual sensor network systems.The text begins with an introduction to passivity and passivity-based control together with the other foundations needed in this book. The main body of the book consists of three parts. The first examines how passivity can be utilized for bilateral teleoperation and demonstrates the inherent robustness of the passivity-based controller against communication delays. The second part emphasizes passivitys usefulness for visual feedback control and estimation. Convergence is rigorously proved even when other passive components are interconnected. The passivity approach is also differentiated from other methodologies. The third part presents the unified passivity-based control-design methodology for multi-agent systems. This scheme is shown to be either immediately applicable or easily extendable to the solution of various motion coordination problems including 3-D attitude/pose synchronization, flocking control and cooperative motion estimation.Academic researchers and practitioners working in systems and control and/or robotics will appreciate the potential of the elegant and novel approach to the control of networked robots presented here. The limited background required and the case-study work described also make the text appropriate for and, it is hoped, inspiring to students.
In this paper, we address a class of distributed optimization problems in the presence of inter-agent communication delays based on passivity. We first focus on unconstrained distributed optimization and provide a passivity-based perspective for distributed optimization algorithms. This perspective allows us to handle communication delays while using scattering transformation. Moreover, we extend the results to constrained distributed optimization, where it is shown that the problem is solved by just adding one more feedback loop of a passive system to the solution of the unconstrained ones. We also show that delays can be incorporated in the same way as the unconstrained problems. Finally, the algorithm is applied to a visual human localization problem using a pedestrian detection algorithm.
In this paper we provide a unifying energy-based approach to the modeling and stability analysis of power systems coupled with market dynamics. We consider a standard model of the power network with a third-order model for the synchronous generators involving voltage dynamics. By applying the primal-dual gradient method to a social welfare problem, a distributed dynamic pricing algorithm is obtained, which can be naturally formulated in port-Hamiltonian form. By interconnection with the physical model a closed-loop port-Hamiltonian system is obtained, whose properties are exploited to prove asymptotic stability to the set of optimal points. This result is extended to the case that also general nodal power constraints are included into the social welfare problem. Additionally, the case of line congestion and power transmission costs in acyclic networks is covered as well. Finally, a dynamic pricing algorithm is proposed that does not require knowledge about the power supply and demand.
This paper deals with a consensus problem for directed two-layered networks in which dynamics of agents are driven by topologies on both cooperative and competitive graphs. A distributed algorithm for exchanging information between agents in the networks is introduced, where the communication noises are taken into account. A sufficient condition for the proposed algorithm to achieve the socalled ε-averaging consensus, i.e., system state gets arbitrarily close to the average value of agents' states with desired high probability, is shown. A rigorous stopping rule for the algorithm is also derived. It is shown that all the agents' states converge to the average value of the initial states when the modeling parameters are chosen appropriately.
This paper studies the asymptotic convergence properties of the primal-dual
dynamics designed for solving constrained concave optimization problems using
classical notions from stability analysis. We motivate the need for this study
by providing an example that rules out the possibility of employing the
invariance principle for hybrid automata to study asymptotic convergence. We
understand the solutions of the primal-dual dynamics in the Caratheodory sense
and characterize their existence, uniqueness, and continuity with respect to
the initial condition. We use the invariance principle for discontinuous
Caratheodory systems to establish that the primal-dual optimizers are globally
asymptotically stable under the primal-dual dynamics and that each solution of
the dynamics converges to an optimizer.
Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.
Delay differential equations, differential integral equations and functional differential equations have been studied for
at least 200 years (see E. Schmitt (1911) for references and some properties of linear equations). Some of the early work
originated from problems in geometry and number theory.
This paper presents an algorithm for compensating delays that are distributed between the sensor(s), controller and actuator(s) within a control loop. This observer-based algorithm is specially suited to compensation of network-induced delays in integrated communication and control systems. The robustness of the algorithm relative to plant model uncertainties has been examined.
We consider a networked control system that utilizes unreliable channels where packets may be lost. By modeling the losses as stochastic processes, we propose a mixed H2/H∞ control approach for synthesis of a controller that depends on the information regarding the losses. The approach is applied to a bilateral teleoperating system. Numerical results show that the proposed control achieves desired performance.
Johansson: A survey of distributed optimization
- T Yang
- X Yi
- J Wu
- Y Yuan
- Z Wu
- Y Meng
- H Hong
- Z Wang
- K H Lin
T. Yang, X. Yi, J. Wu, Y. Yuan, D Wu, Z. Meng, Y. Hong,
H. Wang, Z. Lin, and K.H. Johansson: A survey of distributed
optimization, Annual Reviews in Control, Vol. 47, pp. 278-305,
2019.
ADMM and accelerated ADMM as continuous dynamical systems
- G França
- D P Robinson
- R Vidal
G. França, D.P. Robinson, and R. Vidal: ADMM and accelerated ADMM as continuous dynamical systems, Proceedings
of the 35th International Conference on Machine Learning,
pp. 1559-1567, 2018.