  We consider stochastic optimization problems with functional constraints. If
the objective and constraint functions are not convex, the classical stochastic
approximation algorithms such as the proximal stochastic gradient descent do
not lead to efficient algorithms. In this work, we put forth an accelerated SCA
algorithm that utilizes the recursive momentum-based acceleration which is
widely used in the unconstrained setting. Remarkably, the proposed algorithm
also achieves the optimal SFO complexity, at par with that achieved by
state-of-the-art (unconstrained) stochastic optimization algorithms and match
the SFO-complexity lower bound for minimization of general smooth functions. At
each iteration, the proposed algorithm entails constructing convex surrogates
of the objective and the constraint functions, and solving the resulting convex
optimization problem. A recursive update rule is employed to track the gradient
of the objective function, and contributes to achieving faster convergence and
improved SFO complexity. A key ingredient of the proof is a new parameterized
version of the standard Mangasarian-Fromowitz Constraints Qualification, that
allows us to bound the dual variables and hence establish that the iterates
approach an $\epsilon$-stationary point. We also detail a obstacle-avoiding
trajectory optimization problem that can be solved using the proposed
algorithm, and show that its performance is superior to that of the existing
algorithms. The performance of the proposed algorithm is also compared against
that of a specialized sparse classification algorithm on a binary
classification problem.
