  This report considers the problem of Byzantine fault-tolerance in multi-agent
collaborative optimization. In this problem, each agent has a local cost
function. The goal of a collaborative optimization algorithm is to compute a
minimum of the aggregate of the agents' cost functions. We consider the case
when a certain number of agents may be Byzantine faulty. Such faulty agents may
not follow a prescribed algorithm, and they may send arbitrary or incorrect
information regarding their local cost functions. A reasonable goal in presence
of such faulty agents is to minimize the aggregate cost of the non-faulty
agents. In this report, we show that this goal can be achieved if and only if
the cost functions of the non-faulty agents have a minimal redundancy property.
We present different algorithms that achieve such tolerance against faulty
agents, and demonstrate a trade-off between the complexity of an algorithm and
the properties of the agents' cost functions.
  Further, we also consider the case when the cost functions are independent or
do not satisfy the minimal redundancy property. In that case, we quantify the
tolerance against faulty agents by introducing a metric called weak resilience.
We present an algorithm that attains weak resilience when the faulty agents are
in the minority and the cost functions are non-negative.
