  We study the problem of constrained distributed optimization in multi-agent
networks when some of the computing agents may be faulty. In this problem, the
system goal is to have all the non-faulty agents collectively minimize a global
objective given by weighted average of local cost functions, each of which is
initially known to a non-faulty agent only. In particular, we are interested in
the scenario when the computing agents are connected by an arbitrary directed
communication network, some of the agents may suffer from crash faults or
Byzantine faults, and the estimate of each agent is restricted to lie in a
common constraint set. This problem finds its applications in social computing
and distributed large-scale machine learning.
  The fault-tolerant multi-agent optimization problem was first formulated by
Su and Vaidya, and is solved when the local functions are defined over the
whole real line, and the networks are fully-connected. In this report, we
consider arbitrary directed communication networks and focus on the scenario
where, local estimates at the non-faulty agents are constrained, and only local
communication and minimal memory carried across iterations are allowed. In
particular, we generalize our previous results on fully-connected networks and
unconstrained optimization to arbitrary directed networks and constrained
optimization. As a byproduct, we provide a matrix representation for iterative
approximate crash consensus. The matrix representation allows us to
characterize the convergence rate for crash iterative consensus.
