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