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In this article, we consider a class of decentralized nonsmooth composite optimization problems over undirected graphs. The global optimization problem is to minimize the sum of local objective functions consisting of a Lipschitz-differentiable convex function and two possibly nonsmooth convex functions, one of which contains a bounded linear operator. The goal is to solve the global optimization problem through decentralized computation and communication over a network of agents without a central coordinator. Through using triple proximal splitting operators to deal with the nonsmooth terms, we come up with a novel decentralized algorithm with uncoordinated stepsizes, where the stepsizes with independent upper bounds are also distributed for agents or edges over the communication network. Furthermore, we establish the sublinear convergence rate for the proposed algorithm in terms of the first-order optimality residual in a nonergodic sense. Simulation experiments on a constrained quadratic programming problem and an optimal load-sharing problem are carried out to verify the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we concentrate on dealing with the distributed optimization problem over a directed network, where each unit possesses its own convex cost function and the principal target is to minimize a global cost function (formulated by the average of all local cost functions) while obeying the network connectivity structure. Most of the existing methods, such as push-sum strategy, have eliminated the unbalancedness induced by the directed network via utilizing column-stochastic weights, which may be infeasible if the distributed implementation requires each unit to gain access to (at least) its out-degree information. In contrast, to be suitable for the directed networks with row-stochastic weights, we propose a new directed distributed Nesterov-like gradient tracking algorithm, named as D-DNGT, that incorporates the gradient tracking into the distributed Nesterov method with momentum terms and employs nonuniform step-sizes. D-DNGT extends a number of outstanding consensus algorithms over strongly connected directed networks. The implementation of D-DNGT is straightforward if each unit locally chooses a suitable step-size and privately regulates the weights on information that acquires from in-neighbors. If the largest step-size and the maximum momentum coefficient are positive and small sufficiently, we can prove that D-DNGT converges linearly to the optimal solution provided that the cost functions are smooth and strongly convex. We provide numerical experiments to confirm the findings in this article and contrast D-DNGT with recently proposed distributed optimization approaches. [ABSTRACT FROM AUTHOR]

This article considers a category of constrained convex optimization problems over multiagent networks. The networked agents aim at collaboratively minimizing the sum of all locally known objective functions over a common convex set. Each agent possesses only its local convex function and its state is constrained to a privately known convex set. A novel distributed algorithm is proposed over time-varying unbalanced directed networks based on epigraph form of the original optimization problem and consensus theory. By incorporating the random sleep scheme, the proposed algorithm allows each agent to independently and randomly decide whether to calculate subgradient and take projection at each iteration, which alleviates the cost of subgradient observation. Besides, it neither resorts to doubly stochastic weight matrices (but only row-stochastic) nor the information of the graph sequence to execute. The convergence of the algorithm is explicitly analyzed under conditions that the sequence of time-varying directed graphs is uniformly jointly strongly connected and the subgradients of all local objective functions are bounded over a convex set. The optimization algorithm ensures zero-gap on the expected distance between the estimated value of each agent and the exact optimal solution. The two simulation cases are presented to demonstrate the practicability of the algorithm and correctness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this article, the problem of the distributed convex optimization is investigated, where the target is to collectively minimize a sum of local convex functions over an unbalanced directed multiagent network. Each agent in the network possesses only its private local objective function, and the sum of all local objective functions constitutes the global objective function. We particularly consider the scenario, where the underlying interaction network is strongly connected and the relevant weight matrix is row stochastic. To collectively figure out the optimization problem, a distributed accelerated convergence algorithm where agents utilize uncoordinated step-sizes is presented by incorporating consensus of multiagent networks into distributed inexact gradient tracking technique. Most of the existing methods require all agents to possess the out-degree information of their in-neighbors, which is impractical and hardly inevitable as interpreted in this article. By utilizing the small-gain theorem, we prove that if the maximum step-size is positive and sufficiently small (constrained by a specific upper bound), the proposed algorithm, termed as SGT-FROST, converges geometrically to the optimal solution given that the objective functions are smooth and strongly convex. A certain convergence rate is also shown. Simulations confirm the findings in this article. [ABSTRACT FROM AUTHOR]

This paper studies a class of distributed convex optimization problems by a set of agents in which each agent only has access to its own local convex objective function and the estimate of each agent is restricted to both coupling linear constraint and individual box constraints. Our focus is to devise a distributed primal-dual gradient algorithm for working out the problem over a sequence of time-varying general directed graphs. The communications among agents are assumed to be uniformly strongly connected. A column-stochastic mixing matrix and a fixed step-size are applied in the algorithm which exactly steers all the agents to asymptotically converge to a global optimal solution. Based on the standard strong convexity and the smoothness assumptions of the objective functions, we show that the distributed algorithm is capable of driving the whole network to geometrically converge to an optimal solution of the convex optimization problem only if the step-size does not exceed some upper bound. We also give an explicit analysis for the convergence rate of the proposed optimization algorithm. Simulations on economic dispatch problems and demand response problems in power systems are performed to illustrate the effectiveness of the proposed optimization algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with solving a large category of convex optimization problems using a group of agents, each only being accessible to its individual convex cost function. The optimization problems are modeled as minimizing the sum of all the agents’ cost functions. The communication process between agents is described by a sequence of time-varying yet balanced directed graphs which are assumed to be uniformly strongly connected. Taking into account the fact that the communication channel bandwidth is limited, for each agent we introduce a vector-valued quantizer with finite quantization levels to preprocess the information to be exchanged. We exploit an event-triggered broadcasting technique to guide information exchange, further reducing the communication cost of the network. By jointly designing the dynamic event-triggered encoding–decoding schemes and the event-triggered sampling rules (to analytically determine the sampling time instant sequence for each agent), a distributed subgradient descent algorithm with constrained information exchange is proposed. By selecting the appropriate quantization levels, all the agents’ states asymptotically converge to a consensus value which is also the optimal solution to the optimization problem, without committing saturation of all the quantizers. We find that one bit of information exchange across each connected channel can guarantee that the optimiztion problem can be exactly solved. Theoretical analysis shows that the event-triggered subgradient descent algorithm with constrained data rate of networks converges at the rate of ${O}({\ln t/{\sqrt {t}}})$. We supply a numerical simulation experiment to demonstrate the effectiveness of the proposed algorithm and to validate the correctness of theoretical results. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with the analysis of the second-order locally dynamical consensus of multiagent systems with nonlinear dynamics in the directed networks with arbitrarily fast switching topologies. In our designed framework, the time-varying network topology is constant in each time interval and then randomly jumps to another topology when certain occasional events occur at some random moments. In addition, we further assume that the dwell time of each topology is unknown in advance and the corresponding adjacency weighted matrix is not necessarily nonnegative due to the probable existence of deteriorated communication channels in the underlying interaction network. By the orthogonal decomposition method, the state vector of the resultant error dynamical system can be further decomposed as two transversal components, one of which evolves along the consensus manifold and the other evolves transversally with the consensus manifold. Then, by introducing the generalized matrix measure and by applying the tools of contraction and circle analysis, the second-order locally dynamical consensus of multiagent systems with arbitrarily fast switching directed topologies is theoretically investigated in detail, and some easily verified sufficient conditions are also presented. It is shown that, under sufficiently large coupling strengths, the random switchings can be effectively tolerated and the consensus can be achieved for all agents in the network. Finally, numerical simulation examples are also provided to demonstrate the feasibility and effectiveness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]