  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.
