  We develop asymptotically optimal policies for the multi armed bandit (MAB),
problem, under a cost constraint. This model is applicable in situations where
each sample (or activation) from a population (bandit) incurs a known bandit
dependent cost. Successive samples from each population are iid random
variables with unknown distribution. The objective is to design a feasible
policy for deciding from which population to sample from, so as to maximize the
expected sum of outcomes of $n$ total samples or equivalently to minimize the
regret due to lack on information on sample distributions, For this problem we
consider the class of feasible uniformly fast (f-UF) convergent policies, that
satisfy the cost constraint sample-path wise. We first establish a necessary
asymptotic lower bound for the rate of increase of the regret function of f-UF
policies. Then we construct a class of f-UF policies and provide conditions
under which they are asymptotically optimal within the class of f-UF policies,
achieving this asymptotic lower bound. At the end we provide the explicit form
of such policies for the case in which the unknown distributions are Normal
with unknown means and known variances.
