In this paper, we consider a min-max optimization problem under adversarial manipulation, where there are n cost functions, up to f of which may be replaced by arbitrary faulty functions by an adversary. The goal is to minimize the maximum cost over x among the n functions despite the faulty functions. The problem formulation could naturally extend to Byzantine fault-tolerant distributed min-max optimization. We present a simple algorithm for Byzantine min-max optimization, and provide bounds on the output of the algorithm. We also present an approximate algorithm for this problem. We then extend the problem to a distributed setting and present a distributed algorithm. To the best of our knowledge, we are the first to consider this problem.