  In this paper, we consider constrained optimization problems with convex,
smooth objective and constraints. We propose a new stochastic gradient
algorithm, called the Stochastic Moving Ball Approximation (SMBA) method, to
solve this class of problems, where at each iteration we first take a gradient
step for the objective function and then perform a projection step onto one
ball approximation of a randomly chosen constraint. The computational
simplicity of SMBA, which uses first-order information and considers only one
constraint at a time, makes it suitable for large-scale problems with many
functional constraints. We provide a convergence analysis for the SMBA
algorithm using basic assumptions on the problem, that yields new convergence
rates in both optimality and feasibility criteria evaluated at some average
point. Our convergence proofs are novel since we need to deal properly with
infeasible iterates and with quadratic upper approximations of constraints that
may yield empty balls. We derive convergence rates of order $\mathcal{O}
(k^{-1/2})$ when the objective function is convex, and $\mathcal{O} (k^{-1})$
when the objective function is strongly convex. Preliminary numerical
experiments on quadratically constrained quadratic problems demonstrate the
viability and performance of our method when compared to some existing
state-of-the-art optimization methods and software.
