An optimal algorithm for high-order multi-agent systems with event-triggered communication

The distributed optimal coordinated control problem of heterogeneous linear multi-agent systems is studied in this paper. Specifically, each agent in the system under consideration has a local private cost function, which is assumed to be strongly convex with a Lipschitz continuous gradient. The agents aim to achieve output consensus on the optimal solution of a global cost function, which is the sum of local cost functions. The communication topology among agents is assumed to be a weight-unbalanced and strongly connected directed graph. To solve such a control problem, a tracking-based approach is suggested and utilized, where an auxiliary system is designed to seek the optimal solution. Based on the output regulation technique, distributed tracking control inputs are proposed for the agents, which ensure that the outputs of the agents track the trajectories generated by the auxiliary system. Under the assumptions on the local cost functions, the communication topology and system matrices, it is proved that with the proposed control scheme, the outputs of all agents can achieve consensus on the optimal solution. Finally, two numerical simulation examples are presented to illustrate the effectiveness of the proposed tracking control scheme.

Since the initial proposal in the late 80s, spectral gradient methods continue to receive significant attention, especially due to their excellent numerical performance on various large scale applications. However, to date, they have not been sufficiently explored in the context of distributed optimization. In this paper, we consider unconstrained distributed optimization problems where n nodes constitute an arbitrary connected network and collaboratively minimize the sum of their local convex cost functions. In this setting, building from existing exact distributed gradient methods, we propose a novel exact distributed gradient method wherein nodes’ step-sizes are designed according to the novel rules akin to those in spectral gradient methods. We refer to the proposed method as Distributed Spectral Gradient method. The method exhibits R-linear convergence under standard assumptions for the nodes’ local costs and safeguarding on the algorithm step-sizes. We illustrate the method’s performance through simulation examples.

This article studies the distributed convex optimization problem with a common decision variable, a global inequality constraint, and local constraint sets over a time-varying multiagent network, the objective function of which is a sum of agents’ local convex cost functions. To solve such problem, a penalty-based distributed continuous-time subgradient algorithm with time-varying gain is developed for each agent to seek the saddle point of the penalty Lagrangian function. It is shown that an exact primal optimal solution can be obtained with certain assumption on time-varying gain. Moreover, the proposed algorithm adopts fixed-time projection scheme to ensure that for any initial state value, each local state estimate converges to its convex constraint set within fixed time. Finally, numerical examples are provided to show the effectiveness of the theoretical results.

We consider a min-max optimization problem over a time-varying network of computational agents, where each agent in the network has its local convex cost function which is a private knowledge of the agent. The agents want to jointly minimize the maximum cost incurred by any agent in the network, while maintaining the privacy of their objective functions. To solve the problem, we consider subgradient algorithms where each agent computes its own estimates of an optimal point based on its own cost function, and it communicates these estimates to its neighbors in the network. The algorithms employ techniques from convex optimization, stochastic approximation and averaging protocols (typically used to ensure a proper information diffusion over a network), which allow time-varying network structure. We discuss two algorithms, one based on exact-penalty approach and the other based on primal-dual Lagrangian approach, where both approaches utilize Bregman-distance functions.We establish convergence of the algorithms (with probability one) for a diminishing step-size, and demonstrate the applicability of the algorithms by considering a power allocation problem in a cellular network.

SummaryIn this paper, we study a distributed constrained optimal consensus problem for discrete‐time first‐order integrator systems under bounded input. Each agent is assigned with a local convex cost function, and all agents are required to achieve consensus at the minimum of the aggregate cost over a common convex constraint set, which is only accessible by part of the agents. A 2‐step control protocol is designed to solve the problem under bounded input. Firstly, at each time step, each agent moves a bounded step of the subgradient descent from an individual cost and another one along the projection direction if it can access the constraint. The second movement is then given by a bounded average of the relative positions to neighbors. Specifically, to coordinate the subgradient step within the network without using global information, we introduce an estimate of the upper bound of all agents' subgradients, which is updated by a unilateral consensus mechanism. Under the given control protocol, we obtain the necessary and sufficient conditions to achieve the constrained optimal consensus for a fixed topology. Under similar conditions, we also solve the problem for switching topologies and conduct a convergence rate analysis for strongly convex costs.

On Nash-solvability in pure stationary strategies of finite games with perfect information which may have cycles

This paper studies the distributed generalized Nash equilibrium seeking problem for aggregative games with coupling constraints, where each player optimizes its strategy depending on its local cost function and the estimated strategy aggregation. The information transmission in distributed networks may go beyond bandwidth capacity and eventuate communication bottlenecks. Therefore, we propose a novel communication-efficient distributed generalized Nash equilibrium seeking algorithm, in which the communication efficiency is improved by event-triggered communication and information compression methods. The proposed algorithm saves the transmitted rounds and bits of communication simultaneously. Specifically, by developing precise step size conditions, the proposed algorithm ensures provable convergence, and is proven to achieve (0,δ)-differential privacy with a stochastic quantization scheme. In the end, simulation results verify the effectiveness of the proposed algorithm.

This letter addresses the problem of economic dispatch for network systems where the local cost functions are coupled through their decision variables. Using Lyapunov based techniques, a continuous-time algorithm is developed which guarantees finite-time convergence to the optimal solution. This makes it possible to predetermine the system convergence time based on the initial conditions. Unlike many previous studies, the local cost functions in this letter can be nonconvex and coupled through their decision variables. It is only required that their sum is strongly convex. To verify the effectiveness of the proposed results, an illustrative example is studied.

In this chapter, we focus on introducing a distributed online optimization problem for a set of nodes communicating on a time-varying unbalanced directed network, while considering the problem of how to preserve the privacy of their local cost functions. The main goal of this set of nodes is to cooperatively minimize the sum of all locally known convex cost functions (global cost function). We propose a differentially private distributed stochastic subgradient-push algorithm, named DP-DSSP, to solve such an optimization problem in a collaborative and distributed manner. The algorithm ensures that nodes interact with their in-neighbors and collectively optimize the global cost function. Unlike most existing distributed algorithms that do not consider privacy issues, DP-DSSP successfully protects the privacy of participating nodes through a differential privacy strategy, which is more practical in applications involving sensitive information, such as military affairs or medical treatment. An important feature of DP-DSSP is that it handles distributed online optimization problems in the context of time-varying unbalanced directed networks. Theoretical analysis shows that DP-DSSP can effectively protect differential privacy and can achieve sublinear regret. The tradeoff between the level of privacy and accuracy of DP-DSSP is also revealed. Moreover, DP-DSSP is able to handle arbitrarily large but uniformly bounded delays in the communication link. Finally, simulation experiments confirm the usefulness of DP-DSSP and the findings of this chapter.KeywordsDistributed online optimizationDifferential privacyStochastic subgradient-push algorithmTime-varying directed networksCommunication delays

This paper considers the problem of multi-agent distributed optimization. In this problem, there are multiple agents in the system, and each agent only knows its local cost function. The objective for the agents is to collectively compute a common minimum of the aggregate of all their local cost functions. In principle, this problem is solvable using a distributed variant of the traditional gradient-descent method, which is an iterative method. However, the speed of convergence of the traditional gradient-descent method is highly influenced by the conditioning of the optimization problem being solved. Specifically, the method requires a large number of iterations to converge to a solution if the optimization problem is ill-conditioned. In this paper, we propose an iterative pre-conditioning approach that can significantly attenuate the influence of the problem's conditioning on the convergence-speed of the gradient-descent method. The proposed pre-conditioning approach can be easily implemented in distributed systems and has minimal computation and communication overhead. For now, we only consider a specific distributed optimization problem wherein the individual local cost functions of the agents are quadratic. Besides the theoretical guarantees, the improved convergence speed of our approach is demonstrated through experiments on a real data-set.

This article considers a distributed Nash equilibrium seeking problem, where the players only have partial access to other players' actions, such as their neighbors' actions. Thus, the players are supposed to communicate with each other to estimate other players' actions. To solve the problem, a leader-following consensus gradient-free distributed Nash equilibrium seeking algorithm is proposed. This algorithm utilizes only the measurements of the player' local cost function without the knowledge of its explicit expression or the requirement on its smoothness. Hence, the algorithm is gradient-free during the entire updating process. Moreover, the analysis on the convergence of the Nash equilibrium is studied for the algorithm with both diminishing and constant step-sizes, respectively. Specifically, in the case of diminishing step-size, it is shown that the players' actions converge to the Nash equilibrium almost surely, while in the case of fixed step-size, the convergence to the neighborhood of the Nash equilibrium is achieved. The performance of the proposed algorithm is verified through numerical simulations.

We consider the distributed solution of dynamic programming (DP) problems by policy iteration. We envision a network of processors, each updating asynchronously a local policy and a local cost function, defined on a portion of the state space. The computed values are communicated asynchronously between processors and are used to perform the local policy and cost updates. The natural algorithm of this type can fail even under favorable circumstances, as shown by Williams and Baird [WiB93]. We propose an alternative and almost as simple algorithm, which converges to the optimum under the most general conditions, including asynchronous updating by multiple processors using outdated local cost functions of other processors.

This paper investigates the stochastic distributed nonconvex optimization problem of minimizing a global cost function formed by the summation of n local cost functions. We solve such a problem by involving zeroth-order (ZO) information exchange. In this paper, we propose a ZO distributed primal–dual coordinate method (ZODIAC) to solve the stochastic optimization problem. Agents approximate their own local stochastic ZO oracle along with coordinates under an adaptive smoothing parameter. We show that the proposed algorithm achieves the convergence rate of ${\mathcal{O}}(\sqrt p /\sqrt T )$ for general nonconvex cost functions. We demonstrate the efficiency of proposed algorithms through a numerical example in comparison with the existing state-of-the-art centralized and distributed ZO algorithms.

In this paper, we consider the problem of multi-agent systems where each agent aims to establish a line formation in a distributed manner. In constructing an efficient line formation, finding a line with the closest total distance from every agent is essential. We propose a formation control using least squares estimation (LSE) performed by each agent with only the local information that consists of the corresponding agent’s and neighbors’ positions. Each agent calculates the local cost function, which is the squared distance from the LSE line to the related agent’s and its neighbors’ positions. Our goal is to minimize the global cost function, which is the sum of these local cost functions. To achieve this, we employ distributed optimization to the global cost function of the overall system using the subgradient method performed by each agent locally. We evaluate our proposed method using numerical simulation, and the result complies with our goal of this paper

This paper considers distributed nonconvex optimization with the cost functions being distributed over agents. Noting that information compression is a key tool to reduce the heavy communication load for distributed algorithms as agents iteratively communicate with neighbors, we propose three distributed primal–dual algorithms with compressed communication. The first two algorithms are applicable to a general class of compressors with bounded relative compression error and the third algorithm is suitable for two general classes of compressors with bounded absolute compression error. We show that the proposed distributed algorithms with compressed communication have comparable convergence properties as state-of-the-art algorithms with exact communication. Specifically, we show that they can find first-order stationary points with sublinear convergence rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(1/T)$</tex-math></inline-formula> when each local cost function is smooth, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> is the total number of iterations, and find global optima with linear convergence rate under an additional condition that the global cost function satisfies the Polyak–Łojasiewicz condition. Numerical simulations are provided to illustrate the effectiveness of the theoretical results.