Asynchronous Distributed Optimization with Heterogeneous Regularizations and Normalizations

As multi-agent networks grow in size and scale, they become increasingly difficult to synchronize, though agents must work together even when generating and sharing different information at different times. Targeting such cases, this paper presents an asynchronous optimization framework in which the time between successive communications and computations is unknown and unspecified for each agent. Agents' updates are carried out in blocks, with each agent updating only a small subset of all decision variables. To provide robustness to asynchrony, each agent uses an independently chosen Tikhonov regularization. Convergence is measured with respect to a weighted block-maximum norm in which convergence of agents' blocks can be measured in different p-norms and weighted differently to heterogeneously normalize problems. Asymptotic convergence is shown and convergence rates are derived explicitly in terms of a problem's parameters, with only mild restrictions imposed upon them. Simulation results are provided to verify the theoretical developments made.