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Register a new userDistributed Grid Optimization via Distributed Dual Subgradient Methods with Averaging
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Added by
Dinh Hoa Nguyen
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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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.
Considering the constrained stochastic optimization problem over a time-varying random network, where the agents are to collectively minimize a sum of objective functions subject to a common constraint set, we investigate asymptotic properties of a distributed algorithm based on dual averaging of gradients. Different from most existing works on distributed dual averaging algorithms that mainly concentrating on their non-asymptotic properties, we not only prove almost sure convergence and the rate of almost sure convergence, but also asymptotic normality and asymptotic efficiency of the algorithm. Firstly, for general constrained convex optimization problem distributed over a random network, we prove that almost sure consensus can be archived and the estimates of agents converge to the same optimal point. For the case of linear constrained convex optimization, we show that the mirror map of the averaged dual sequence identifies the active constraints of the optimal solution with probability 1, which helps us to prove the almost sure convergence rate and then establish asymptotic normality of the algorithm. Furthermore, we also verify that the algorithm is asymptotically optimal. To the best of our knowledge, it seems to be the first asymptotic normality result for constrained distributed optimization algorithms. Finally, a numerical example is provided to justify the theoretical analysis.
Optimization in distributed networks plays a central role in almost all distributed machine learning problems. In principle, the use of distributed task allocation has reduced the computational time, allowing better response rates and higher data reliability. However, for these computational algorithms to run effectively in complex distributed systems, the algorithms ought to compensate for communication asynchrony, network node failures and delays known as stragglers. These issues can change the effective connection topology of the network, which may vary over time, thus hindering the optimization process. In this paper, we propose a new distributed unconstrained optimization algorithm for minimizing a convex function which is adaptable to a parameter server network. In particular, the network worker nodes solve their local optimization problems, allowing the computation of their local coded gradients, which will be sent to different server nodes. Then within this parameter server platform each server node aggregates its communicated local gradients, allowing convergence to the desired optimizer. This algorithm is robust to network s worker node failures, disconnection, or delaying nodes known as stragglers. One way to overcome the straggler problem is to allow coding over the network. We further extend this coding framework to enhance the convergence of the proposed algorithm under such varying network topologies. By using coding and utilizing evaluations of gradients of uniformly bounded delay we further enhance the proposed algorithm performance. Finally, we implement the proposed scheme in MATLAB and provide comparative results demonstrating the effectiveness of the proposed framework
Distributed optimization is concerned with using local computation and communication to realize a global aim of optimizing the sum of local objective functions. It has gained wide attention for a variety of applications in networked systems. This paper addresses a class of constrained distributed nonconvex optimization problems involving univariate objective functions, aiming to achieve global optimization without requiring local evaluations of gradients at every iteration. We propose a novel algorithm named CPCA, exploiting the notion of combining Chebyshev polynomial approximation, average consensus and polynomial optimization. The proposed algorithm is i) able to obtain $epsilon$ globally optimal solutions for any arbitrarily small given accuracy $epsilon$, ii) efficient in terms of both zeroth-order queries (i.e., evaluations of function values) and inter-agent communication, and iii) distributed terminable when the specified precision requirement is met. The key insight is to use polynomial approximations to substitute for general objective functions, and turn to solve an easier approximate version of the original problem. Due to the nice analytic properties owned by polynomials, this approximation not only facilitates efficient global optimization, but also allows the design of gradient-free iterations to reduce cumulative costs of queries and achieve geometric convergence when nonconvex problems are solved. We provide comprehensive analysis of the accuracy and complexities of the proposed algorithm.
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