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This paper proposes a set of novel optimization algorithms for solving a class of convex optimization problems with time-varying streaming cost functions. We develop an approach to track the optimal solution with a bounded error. Unlike prior work, our algorithm is executed only by using the first-order derivatives of the cost function, which makes it computationally efficient for optimization with time-varying cost function. We compare our algorithms to the gradient descent algorithm and show why gradient descent is not an effective solution for optimization problems with time-varying cost. Several examples, including solving a model predictive control problem cast as a convex optimization problem with a streaming time-varying cost function, demonstrate our results.more » « lessFree, publicly-accessible full text available July 10, 2025
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Free, publicly-accessible full text available June 1, 2025
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A Study of Privacy Preservation in Average Consensus Algorithm via Deterministic Obfuscation SignalsThis article is a study on the use of additive obfuscation signals to keep the reference values of the agents in the continuous-time Laplacian average consensus algorithm private from eavesdroppers. Obfuscation signals are perturbations that agents add to their local dynamics and their transmitted-out messages to conceal their private reference values. An eavesdropper is an agent inside or outside the network that has access to some subset of the interagent communication messages, and its knowledge set also includes the network topology. Rather than focusing on using a zero-sum and vanishing additive signal, our work determines the necessary and sufficient conditions that define the set of admissible obfuscation signals that do not perturb the convergence point of the algorithm from the average of the reference values of the agents. Of theoretical interest, our results show that this class includes nonvanishing signals as well. Given this broader class of admissible obfuscation signals, we define a deterministic notion of privacy preservation. In this definition, privacy preservation for an agent means that neither the private reference value nor a finite set of values to which the private reference value of the agent belongs to can be obtained. Then, we evaluate the agents’ privacy against eavesdroppers with different knowledge sets.more » « lessFree, publicly-accessible full text available March 1, 2025
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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.more » « less
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We consider an in-network optimal resource allocation problem in which a group of agents interacting over a connected graph want to meet a demand while minimizing their collective cost. The contribution of this paper is to design a distributed continuous-time algorithm for this problem inspired by a recently developed first-order transformed primal-dual method. The solution applies to cluster-based setting where each agent may have a set of subagents, and its local cost is the sum of the cost of these subagents. The proposed algorithm guarantees an exponential convergence for strongly convex costs and asymptotic convergence for convex costs. Exponential convergence when the local cost functions are strongly convex is achieved even when the local gradients are only locally Lipschitz. For convex local cost functions, our algorithm guarantees asymptotic convergence to a point in the minimizer set. Through numerical examples, we show that our proposed algorithm delivers a faster convergence compared to existing distributed resource allocation algorithms.more » « less
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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.more » « less