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Register a new userConvergence Properties of the Distributed Projected Subgradient Algorithm over General Graphs
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In this paper, we revisit a well-known distributed projected subgradient algorithm which aims to minimize a sum of cost functions with a common set constraint. In contrast to most of existing results, weight matrices of the time-varying multi-agent network are assumed to be more general, i.e., they are only required to be row stochastic instead of doubly stochastic. We focus on analyzing convergence properties of this algorithm under general graphs. We first show that there generally exists a graph sequence such that the algorithm is not convergent when the network switches freely within finitely many general graphs. Then to guarantee the convergence of this algorithm under any uniformly jointly strongly connected general graph sequence, we provide a necessary and sufficient condition, i.e., the intersection of optimal solution sets to all local optimization problems is not empty. Furthermore, we surprisingly find that the algorithm is convergent for any periodically switching general graph sequence, and the converged solution minimizes a weighted sum of local cost functions, where the weights depend on the Perron vectors of some product matrices of the underlying periodically switching graphs. Finally, we consider a slightly broader class of quasi-periodically switching graph sequences, and show that the algorithm is convergent for any quasi-periodic graph sequence if and only if the network switches between only two graphs.
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We investigate a distributed optimization problem over a cooperative multi-agent time-varying network, where each agent has its own decision variables that should be set so as to minimize its individual objective subject to local constraints and global coupling constraints. Based on push-sum protocol and dual decomposition, we design a distributed regularized dual gradient algorithm to solve this problem, in which the algorithm is implemented in time-varying directed graphs only requiring the column stochasticity of communication matrices. By augmenting the corresponding Lagrangian function with a quadratic regularization term, we first obtain the bound of the Lagrangian multipliers which does not require constructing a compact set containing the dual optimal set when compared with most of primal-dual based methods. Then, we obtain that the convergence rate of the proposed method can achieve the order of $mathcal{O}(ln T/T)$ for strongly convex objective functions, where $T$ is the iterations. Moreover, the explicit bound of constraint violations is also given. Finally, numerical results on the network utility maximum problem are used to demonstrate the efficiency of the proposed algorithm.
A stochastic incremental subgradient algorithm for the minimization of a sum of convex functions is introduced. The method sequentially uses partial subgradient information and the sequence of partial subgradients is determined by a general Markov chain. This makes it suitable to be used in networks where the path of information flow is stochastically selected. We prove convergence of the algorithm to a weighted objective function where the weights are given by the Ces`aro limiting probability distribution of the Markov chain. Unlike previous works in the literature, the Ces`aro limiting distribution is general (not necessarily uniform), allowing for general weighted objective functions and flexibility in the method.
A collection of optimization problems central to power system operation requires distributed solution architectures to avoid the need for aggregation of all information at a central location. In this paper, we study distributed dual subgradient methods to solve three such optimization problems. Namely, these are tie-line scheduling in multi-area power systems, coordination of distributed energy resources in radial distribution networks, and joint dispatch of transmission and distribution assets. With suitable relaxations or approximations of the power flow equations, all three problems can be reduced to a multi-agent constrained convex optimization problem. We utilize a constant step-size dual subgradient method with averaging on these problems. For this algorithm, we provide a convergence guarantee that is shown to be order-optimal. We illustrate its application on the grid optimization problems.
This paper studies the distributed optimization problem where the objective functions might be nondifferentiable and subject to heterogeneous set constraints. Unlike existing subgradient methods, we focus on the case when the exact subgradients of the local objective functions can not be accessed by the agents. To solve this problem, we propose a projected primal-dual dynamics using only the objective functions approximate subgradients. We first prove that the formulated optimization problem can only be solved with an approximate error depending upon the accuracy of the available subgradients. Then, we show the exact solvability of this optimization problem if the accumulated approximation error is not too large. After that, we also give a novel componentwise normalized variant to improve the transient behavior of the convergent sequence. The effectiveness of our algorithms is verified by a numerical example.
Dual decomposition is widely utilized in distributed optimization of multi-agent systems. In practice, the dual decomposition algorithm is desired to admit an asynchronous implementation due to imperfect communication, such as time delay and packet drop. In addition, computational errors also exist when individual agents solve their own subproblems. In this paper, we analyze the convergence of the dual decomposition algorithm in distributed optimization when both the asynchrony in communication and the inexactness in solving subproblems exist. We find that the interaction between asynchrony and inexactness slows down the convergence rate from $mathcal{O} ( 1 / k )$ to $mathcal{O} ( 1 / sqrt{k} )$. Specifically, with a constant step size, the value of objective function converges to a neighborhood of the optimal value, and the solution converges to a neighborhood of the exact optimal solution. Moreover, the violation of the constraints diminishes in $mathcal{O} ( 1 / sqrt{k} )$. Our result generalizes and unifies the existing ones that only consider either asynchrony or inexactness. Finally, numerical simulations validate the theoretical results.
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