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Noise-to-state exponentially stable distributed convex optimization on weight-balanced digraphs
D. Mateos-Nuñez, J. Cortés
SIAM Journal on Control and Optimization 54 (1) (2016), 266-290

Abstract

This paper studies the robustness properties against additive persistent noise of a class of distributed continuous-time algorithms for convex optimization. A team of agents, each with its own private objective function, seeks to collectively determine the global decision vector that minimizes the sum of the objective functions. The team communicates over a weight-balanced, strongly connected digraph and both inter-agent communication and agent computation are corrupted by noise. Under the proposed distributed algorithm, each agent updates its estimate of the global solution using the gradient of its local objective function while, at the same time, seeking to agree with its neighbors' estimates via proportional-integral feedback on their disagreement. Under mild conditions on the local objective functions, we show that this strategy is noise-to-state exponentially stable in second moment with respect to the optimal solution. Our technical approach combines notions and tools from graph theory, stochastic differential equations, Lyapunov stability analysis, and co-coercivity of vector fields. Simulations illustrate our results.

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