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This paper proposes a novel solution for the distributed unconstrained optimization problem where the total cost is the summation of time-varying local cost functions of a group networked agents. The objective is to track the optimal trajectory that minimizes the total cost at each time instant. Our approach consists of a two-stage dynamics, where the first one samples the first and second derivatives of the local costs periodically to construct an estimate of the descent direction towards the optimal trajectory, and the second one uses this estimate and a consensus term to drive local states towards the time-varying solution while reaching consensus. The first part is carried out by a weighted average consensus algorithm in the discrete-time framework and the second part is performed with a continuous-time dynamics. Using the Lyapunov stability analysis, an upper bound on the gradient of the total cost is obtained which is asymptotically reached. This bound is characterized by the properties of the local costs. To demonstrate the performance of the proposed method, a numerical example is conducted that studies tuning the algorithm’s parameters and their effects on the convergence of local states to the optimal trajectory. 

This paper proposes a distributed solution for an optimal resource allocation problem with a time-varying cost function and time-varying demand. The objective is to minimize a global cost, which is the summation of local quadratic time-varying cost functions, by allocating time-varying resources. A reformulation of the original problem is developed and is solved in a distributed manner using only local interactions over an undirected connected graph. In the proposed algorithm, the local state trajectories converge to a bounded neighborhood of the optimal trajectory. This bound is characterized in terms the parameters of the cost and topology properties. We also show that despite the tracking error, the trajectories are feasible at all times, meaning that the resource allocation equality constraint is met at every execution time. Our algorithm also considers the possibility of some generators going out of production from time to time and adjusts the solution so that the remaining generators can meet the demands in an optimal manner. Numerical examples demonstrate our results. 

A critical factor for expanding the adoption of networked solutions is ensuring local data privacy of in-network agents implementing a distributed algorithm. In this paper, we consider privacy preservation in the distributed optimization problem in the sense that local cost parameters should not be revealed. Current approaches to privacy preservation normally propose methods that sacrifice exact convergence or increase communication overhead. We propose PrivOpt, an intrinsically private distributed optimization algorithm that converges exponentially fast without any convergence error or using extra communication channels. We show that when the number of the parameters of the local cost is greater than the dimension of the decision variable of the problem, no malicious agent, even if it has access to all transmitted-in and -out messages in the network, can obtain local cost parameters of other agents. As an application study, we show how our proposed PrivOpt algorithm can be used to solve an optimal resource allocation problem with the guarantees that the local cost parameters of all the agents stay private.