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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.
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Distributed Stochastic Optimization with Gradient Tracking over Time- Varying Directed Networks
We study a distributed method called SAB–TV, which employs gradient tracking to collaboratively minimize the strongly-convex sum of smooth local cost functions for networked agents communicating over a time-varying directed graph. Each agent, assumed to have access to a stochastic first order oracle for obtaining an unbiased estimate of the gradient of its local cost function, maintains an auxiliary variable to asymptotically track the stochastic gradient of the global cost. The optimal decision and gradient tracking are updated over time through limited information exchange with local neighbors using row- and column-stochastic weights, guaranteeing both consensus and optimality. With a sufficiently small constant step-size, we demonstrate that, in expectation, SAB–TV converges linearly to a neighborhood of the optimal solution. Numerical simulations illustrate the effectiveness of the proposed algorithm.
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- Award ID(s):
- 2106336
- PAR ID:
- 10511517
- Publisher / Repository:
- IEEE
- Date Published:
- ISBN:
- 979-8-3503-2574-4
- Page Range / eLocation ID:
- 1605 to 1609
- Format(s):
- Medium: X
- Location:
- Pacific Grove, CA, USA
- Sponsoring Org:
- National Science Foundation
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