This article studies the convex optimization problem with general constraints, where its global objective function is composed of the sum of local objective functions. The objective is to design a distributed algorithm to cooperatively resolve the optimization problem under the condition that only the information of each node’s own local cost function and its neighbors’ states can be obtained. To this end, the optimality condition of the researched optimization problem is developed in terms of the saddle point theory. On this basis, the corresponding continuous-time primal-dual algorithm is constructed for the considered constrained convex optimization problem under time-varying undirected and connected graphs. In the case that the parameters involved in the proposed algorithm satisfy certain inequality, the states of all nodes will reach consensus in finite time. Meanwhile, the average state is globally convergent to the optimal solution of the considered optimization problem under some mild and standard assumptions.

In this article, the constrained optimization problem with its global objective function being the sum of convex local cost functions and the constraint being a closed convex set is researched. The aim of this study is to solve the researched problem in a distributed manner, that is, using only local computations and local information exchanges. Toward this end, two gradient-tracking-based distributed optimization algorithms are designed for the considered problem over weight-balanced and weight-unbalanced graphs, respectively. Since the classical projection method is unsuitable to handle the closed convex set constraint under the gradient-tracking framework, a new indirect projection method is employed in this article to deal with the involved closed convex set constraint. Furthermore, two time scales are introduced to complete the convergence analyses. In addition, under the condition that all local cost functions are strongly convex and L-smooth, it is proved that the algorithms with well-selected fixed step sizes have linear convergence rates.

This article investigates the convex optimization problem with general local constraints including N equality constraints, N inequality constraints, and N closed convex set constraints. Especially, the objective function of the considered problem is the sum of N convex functions, which are seen as local objective functions. First, for the distributed setting, a weight-unbalanced digraph with N nodes is introduced to describe the communication topology. Clearly, in the distributed setting, each node is responsible for a local objective function accompanied with an equality constraint, an inequality constraint, and a closed convex set constraint, which are viewed as local constraints. The aim of this article is to solve the considered convex optimization problem in a distributed manner, i.e., to lead the states of all nodes to a common optimal solution of the considered problem under the condition that each node only knows its own local objective function, and local constraints, and the information of its neighbors' states. Towards this end, by resorting to the exact penalty function method and the gradient descent-like method, two distinct distributed discrete-time algorithms are developed for two different cases, respectively. Furthermore, the convergence properties of the designed algorithms are rigorously analyzed under some common and standard assumptions, and the convergence rates are described in detail. Finally, simulation results are provided to verify the theoretical results.

This paper is devoted to addressing the distributed aggregative optimization (DAO) problem via compressed gradient tracking algorithms, where the cost function of each agent relies on the aggregation of other agents' decisions as well as its own decision. To this end, a new kind of distributed aggregative gradient tracking algorithm with compression communication is developed based upon the gradient tracking algorithm and the technique of communication compression. Under the scenario with a time-invariant and balanced graph, it is theoretically shown that the present algorithm owns a linear convergence rate (in the mean square error sense) with the strongly convex and smooth cost functions. Furthermore, the result is extended to a more general case with the time-varying graphs considered. Specifically, it is proven that the developed algorithm could converge linearly to the optimal solution of the DAO problem (in the mean square error sense) if the time-varying balanced graph is jointly strongly connected and some suitable conditions are satisfied. With the optimal placement problem considered, some numerical simulation results are performed to validate the theoretical results.

The convergence rate and applicability to directed graphs with interaction topologies are two important features for practical applications of distributed optimization algorithms. In this article, a new kind of fast distributed discrete-time algorithms is developed for solving convex optimization problems with closed convex set constraints over directed interaction networks. Under the gradient tracking framework, two distributed algorithms are, respectively, designed over balanced and unbalanced graphs, where momentum terms and two time-scales are involved. Furthermore, it is demonstrated that the designed distributed algorithms attain linear speedup convergence rates provided that the momentum coefficients and the step size are appropriately selected. Finally, numerical simulations verify the effectiveness and the global accelerated effect of the designed algorithms.

In this paper, a special class of distributed stochastic nonconvex optimization problem is investigated. Each agent in the network only has access to a local stochastic weakly convex objective function and can only communicate with its neighbours in a random time-delay environment. For solving the considered problem with the distributed communication under time-delay interference, the distributed weakly convex delay-tolerance algorithm (DWDTA) with diminishing step-size and fixed step-size are proposed, respectively. Specifically, we show the convergence of the DWDTA with diminishing step-size by using Moreau Envelope measurement and demonstrate the linear convergence of the DWDTA with fixed step-size under sharpness condition. Our convergence results explicitly characterize the influences of the weakly convex function and the random time-delay interference on the convergence performances, respectively. Finally, numerical results are worked out to verify the effectiveness of the DWDTA.