  The problem of distributed optimization requires a group of agents to reach
agreement on a parameter that minimizes the average of their local cost
functions using information received from their neighbors. While there are a
variety of distributed optimization algorithms that can solve this problem,
they are typically vulnerable to malicious (or ``Byzantine'') agents that do
not follow the algorithm. Recent attempts to address this issue focus on single
dimensional functions, or provide analysis under certain assumptions on the
statistical properties of the functions at the agents. In this paper, we
propose a resilient distributed optimization algorithm for multi-dimensional
convex functions. Our scheme involves two filtering steps at each iteration of
the algorithm: (1) distance-based and (2) component-wise removal of extreme
states. We show that this algorithm can mitigate the impact of up to $F$
Byzantine agents in the neighborhood of each regular node, without knowing the
identities of the Byzantine agents in advance. In particular, we show that if
the network topology satisfies certain conditions, all of the regular states
are guaranteed to asymptotically converge to a bounded region that contains the
global minimizer.
