  One of the most important problems in the field of distributed optimization
is the problem of minimizing a sum of local convex objective functions over a
networked system. Most of the existing work in this area focus on developing
distributed algorithms in a synchronous setting under the presence of a central
clock, where the agents need to wait for the slowest one to finish the update,
before proceeding to the next iterate. Asynchronous distributed algorithms
remove the need for a central coordinator, reduce the synchronization wait, and
allow some agents to compute faster and execute more iterations. In the
asynchronous setting, the only known algorithms for solving this problem could
achieve either linear or sublinear rate of convergence. In this work, we built
upon the existing literature to develop and analyze an asynchronous
Newton-based method to solve a penalized version of the problem. We show that
this algorithm guarantees almost sure convergence with global linear and local
quadratic rate in expectation. Numerical studies confirm superior performance
of our algorithm against other asynchronous methods.
