Online Distributed Convex Optimization for Unbalanced Varying Graphs With Delayed Feedback

Feedback signal delays present a common challenge in the online decision-making processes of various real-world systems, such as real-time economic dispatch in power systems. These delays, primarily induced by limited computational capabilities or unknown parameter changes, can significantly hinder the performance or even effectiveness of online decision-making strategies. This study investigates distributed optimization over time-varying unbalanced networks with delayed feedback signals. We propose discrete-time distributed online algorithms for both constrained and unconstrained optimization problems. Each node in the network has access only to its individual time-varying local objective function with a delayed time sequence. At each decision-making step, each node selects the appropriate local online behavior, which depends on the information from its neighbor nodes, towards the minimization of global cumulative cost function value. We demonstrate that sublinear regrets can be ensured as long as the time-varying unbalanced communication networks maintain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {B}$</tex-math> </inline-formula>-strong connectivity. For a general convex or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu $</tex-math> </inline-formula>-strongly convex local objective function, the network regret and individual regret grow as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}((\ln (T))^{2})$</tex-math> </inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(\sqrt {T})$</tex-math> </inline-formula>, respectively, and are intrinsically related to the feedback delays and step sizes. To validate the proposed algorithms, numerical studies are carried out on constrained distributed time-varying economic dispatch problems.