  Anomaly detection in large populations is a challenging but highly relevant
problem. The problem is essentially a multi-hypothesis problem, with a
hypothesis for every division of the systems into normal and anomal systems.
The number of hypothesis grows rapidly with the number of systems and
approximate solutions become a necessity for any problems of practical
interests. In the current paper we take an optimization approach to this
multi-hypothesis problem. We first observe that the problem is equivalent to a
non-convex combinatorial optimization problem. We then relax the problem to a
convex problem that can be solved distributively on the systems and that stays
computationally tractable as the number of systems increase. An interesting
property of the proposed method is that it can under certain conditions be
shown to give exactly the same result as the combinatorial multi-hypothesis
problem and the relaxation is hence tight.
