  In this paper we consider convex optimization problems with stochastic
composite objective function subject to (possibly) infinite intersection of
constraints. The objective function is expressed in terms of expectation
operator over a sum of two terms satisfying a stochastic bounded gradient
condition, with or without strong convexity type properties. In contrast to the
classical approach, where the constraints are usually represented as
intersection of simple sets, in this paper we consider that each constraint set
is given as the level set of a convex but not necessarily differentiable
function. Based on the flexibility offered by our general optimization model we
consider a stochastic subgradient method with random feasibility updates. At
each iteration, our algorithm takes a stochastic proximal (sub)gradient step
aimed at minimizing the objective function and then a subsequent subgradient
step minimizing the feasibility violation of the observed random constraint. We
analyze the convergence behavior of the proposed algorithm for diminishing
stepsizes and for the case when the objective function is convex or strongly
convex, unifying the nonsmooth and smooth cases. We prove sublinear convergence
rates for this stochastic subgradient algorithm, which are known to be optimal
for subgradient methods on this class of problems. When the objective function
has a linear least-square form and the constraints are polyhedral, it is shown
that the algorithm converges linearly. Numerical evidence supports the
effectiveness of our method in real problems.
