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/√t). We supply a numerical simulation experiment to demonstrate the effectiveness of the proposed algorithm and to validate the correctness of theoretical results.

This paper is concerned with a general class of distributed constrained optimization problems over a multi-agent network, where the global objective function is represented by the sum of all local objective functions. Each agent in the network only knows its own local objective function, and is restricted to a global non-empty closed convex set. We discuss the scenario where the communication of the whole multi-agent network is expressed as a sequence of time-varying general unbalanced directed graphs. The directed graphs are required to be uniformly jointly strongly connected and the weight matrices are only rowstochastic. To collaboratively deal with the optimization problems, existing distributed methods mostly require the communication graph to be fixed or balanced, which is impractical and hardly inevitable. In contrast, we propose a new distributed projection subgradient algorithm which is applicable to the time-varying general unbalanced directed graphs and does not need each agent to know its in-neighbors' out-degree. When the objective functions are convex and Lipschitz continuous, it is proved that the proposed algorithm exactly converges to the optimal s

In this paper, the economic dispatch problem (EDP) in smart grids is investigated over a directed network, which concentrates on allocating the generation power among the generators to satisfy the load demands with minimal total generation cost while complying with all constraints of local generation capacity. Each generator possesses its own local generation cost, and the total generation cost is the sum of all local generation costs. To deal with EDP, most of the existing methods, such as strategy based on push-sum, surmount the unbalancedness induced by the directed network via employing column-stochastic weights, which might be infeasible in distributed implementation. In contrast, in order to be suitable for the directed network with row-stochastic weights, we develop a novel directed distributed Lagrangian momentum algorithm, named as D-DLM, which integrates distributed gradient tracking method with two types of momentum terms and utilizes non-uniform step-sizes with respect to the updates of Lagrangian multipliers. If the largest step-size and the maximum momentum coefficient are positive and sufficiently small, D-DLM can linearly allocate the optimal dispatch on condition that the generation costs are smooth and strongly convex. Finally, a variety of studies on EDP in smart grids are simulated.

In this paper, we discuss a class of distributed constrained optimization problems in power systems where the target is to optimize the sum of all agents' local convex objective functions over a general unbalanced directed communication network. Each local convex objective function is known exclusively to a single agent, and the agents' variables are constrained to global coupling linear constraint and individual box constraints. To collaboratively solve the optimization problems, existing distributed methods mostly require the communication network to be balanced or have the knowledge of in-neighbors' out-degree for all agents, which are quite restrictive and hardly inevitable in practical applications. In contrast, we investigate a novel distributed primal-dual augmented (sub)gradient algorithm which utilizes a row-stochastic matrix (does not need each agent to know its in-neighbors out-degree) and employs uncoordinated step-sizes, and yet exactly converges to the optimal solution over a general unbalanced directed communication network. Under the assumptions of the strong convexity and smoothness on the aggregate objective functions, it is proved that the algorithm geometrically converges to the optimal solution if the uncoordinated step-sizes do not exceed the upper bound. An explicit analysis for the convergence rate of the proposed algorithm is also characterized. To manifest effectiveness and applicability of the proposed algorithm, three case studies are presented to solve two practical problems in power systems.