Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method

Abstract

Various distributed optimization methods have been developed for solving problems which have simple local constraint sets and whose objective function is the sum of local cost functions of distributed agents in a network. Motivated by emerging applications in smart grid and distributed sparse regression, this paper studies distributed optimization methods for solving general problems which have a coupled global cost function and have inequality constraints. We consider a network scenario where each agent has no global knowledge and can access only its local mapping and constraint functions. To solve this problem in a distributed manner, we propose a consensus-based distributed primal-dual perturbation (PDP) algorithm. In the algorithm, agents employ the average consensus technique to estimate the global cost and constraint functions via exchanging messages with neighbors, and meanwhile use a local primal-dual perturbed subgradient method to approach a global optimum. The proposed PDP method not only can handle smooth inequality constraints but also non-smooth constraints such as some sparsity promoting constraints arising in sparse optimization. We prove that the proposed PDP algorithm converges to an optimal primal-dual solution of the original problem, under standard problem and network assumptions. Numerical results illustrating the performance of the proposed algorithm for a distributed demand response control problem in smart grid are also presented. DRAFT Distributed optimization methods are becoming popular options for solving several engineering problems, including parameter estimation, detection and localization problems in sensor networks [1], [2], resource allocation problems in peer-to-peer/multi-cellular communication networks [3], [4], and distributed learning and regression problems in control [5] and machine learning [6]-[8],to name a few. In these applications, rather than pooling together all the relevant parameters that define the optimization problem, distributed agents, which have access to a local subset of such parameters, collaborate with each other to minimize a global cost function, subject to local variable constraints. Specifically, since it is not always efficient for the agents to exchange across the network the local cost and constraint functions, owing to the large size of network, time-varying network topology, energy constraints and/or privacy issues, distributed optimization methods that utilize only local information and messages exchanged between connecting neighbors have been of great interest; see [9]-[16] and references therein.Contributions: Different from the existing works [9]- [14] where the local variable constraints are usually simple (in the sense that they can be handled via simple projection) and independent among agents, in this paper, we consider a problem formulation that has a general set of convex inequality constraints that couple all the agents' optimization variables. In addition, similar to [17], the considered problem has a global (non-separable) convex c...

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“…In [27], energy storage is optimally controlled through ICC. In [28], a primal-dual perturbed sub-gradient method is applied locally while averaging consensus is applied to estimate the global cost functions and constaints.…”

A Distributed Optimization Algorithm for the Predictive Control of Smart Grids

“…A notable recent exception is [10], where the global cost function is not separable. In [10], it is assumed that each agent knows the global cost function, but only has access to its local decision variables and local constraint set.…”

“…In [10], it is assumed that each agent knows the global cost function, but only has access to its local decision variables and local constraint set. Furthermore, [10] assumes a global coupled inequality constraint, where each agent knows its (functional) contribution to the global coupled constraint. In this setting, [10] presents a distributed optimization algorithm based on neighbor-to-neighbor communication.…”

“…Furthermore, [10] assumes a global coupled inequality constraint, where each agent knows its (functional) contribution to the global coupled constraint. In this setting, [10] presents a distributed optimization algorithm based on neighbor-to-neighbor communication. Importantly, it is shown that this distributed optimization algorithm converges to the solution of the global optimization problem.…”

“…In contrast to [10], the algorithm presented herein relies on the presence of a central entity with which all agents communicate. Note that this is the only communication present; i.e., agents do not communicate directly with each other.…”