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Abstract. We consider a distributed multi-agent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own function and by projecting onto the constraint set.The goal of this paper is to explore the effects of stochastic subgradient errors on the convergence of the algorithm. We first consider the behavior of the algorithm in mean, and then the convergence with probability 1 and in mean square. We consider general stochastic errors that have uniformly bounded second moments and obtain bounds on the limiting performance of the algorithm in mean for diminishing and non-diminishing stepsizes. When the means of the errors diminish, we prove that there is mean consensus between the agents and mean convergence to the optimum function value for diminishing stepsizes. When the mean errors diminish sufficiently fast, we strengthen the results to consensus and convergence of the iterates to an optimal solution with probability 1 and in mean square.Key words. Distributed algorithm, convex optimization, subgradient methods, stochastic approximation.AMS subject classifications. 90C151. Introduction. A number of problems that arise in the context of wired and wireless networks can be posed as the minimization of a sum of functions, when each component function is available only to a specific agent [23,25,26]. Often, it is not efficient, or not possible, for the network agents to share their objective functions with each other or with a central coordinator. In such scenarios, distributed algorithms that only require the agents to locally exchange limited and high level information are preferable. For example, in a large wireless network, energy is a scarce resource and it might not be efficient for a central coordinator to learn the individual objective functions from each and every agent [23]. In a network of databases from which information is to be mined, privacy considerations may not allow the sharing of the objective functions [34]. In a distributed network on a single chip, for the chip to be fault tolerant, it is desirable to perform the processing in a distributed manner to account for the statistical process variations [32].We consider constrained minimization of a sum of convex functions, where each component function is known partially (with stochastic errors) to a specific network agent. The algorithm proposed builds on the distributed algorithm proposed in [19] for the unconstrained minimization problem. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent averages the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own func...

We consider the uplink power control problem where mobile users in different cells are communicating with their base stations. We formulate the power control problem as the minimization of a sum of convex functions. Each component function depends on the channel coefficients from all the mobile users to a specific base station and is assumed to be known only to that base station (only CSIR). We then view the power control problem as a distributed optimization problem that is to be solved by the base stations and propose convergent, distributed and iterative power control algorithms. These algorithms require each base station to communicate with the base stations in its neighboring cells in each iteration and are hence nonautonomous. Since the base stations are connected through a wired backbone the communication overhead is not an issue. The convergence of the algorithms is shown theoretically and also verified through numerical simulations.

We consider a network of sensors deployed to sense a spatio-temporal field and estimate a parameter of interest. We are interested in the case where the temporal process sensed by each sensor can be modeled as a state-space process that is perturbed by random noise and parametrized by an unknown parameter.To estimate the unknown parameter from the measurements that the sensors sequentially collect, we propose a distributed and recursive estimation algorithm, which we refer to as the incremental recursive prediction error algorithm. This algorithm has the distributed property of incremental gradient algorithms and the on-line property of recursive prediction error algorithms. We study the convergence behavior of the algorithm and provide sufficient conditions for its convergence. Our convergence result is rather general and contains as special cases the known convergence results for the incremental versions of the least-mean square algorithm. Finally, we use the algorithm developed in this paper to identify the source of a gas-leak (diffusing source) in a closed warehouse and also report some numerical results.

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