  This paper considers stochastic convex optimization problems with two sets of
constraints: (a) deterministic constraints on the domain of the optimization
variable, which are difficult to project onto; and (b) deterministic or
stochastic constraints that admit efficient projection. Problems of this form
arise frequently in the context of semidefinite programming as well as when
various NP-hard problems are solved approximately via semidefinite relaxation.
Since projection onto the first set of constraints is difficult, it becomes
necessary to explore projection-free algorithms, such as the stochastic
Frank-Wolfe (FW) algorithm. On the other hand, the second set of constraints
cannot be handled in the same way, and must be incorporated as an indicator
function within the objective function, thereby complicating the application of
FW methods. Similar problems have been studied before; however, they suffer
from slow convergence rates. This work, equipped with momentum based gradient
tracking technique, guarantees fast convergence rates on par with the
best-known rates for problems without the second set of constraints.
Zeroth-order variants of the proposed algorithms are also developed and again
improve upon the state-of-the-art rate results. We further propose the novel
trimmed FW variants that enjoy the same convergence rates as their classical
counterparts, but are empirically shown to require significantly fewer calls to
the linear minimization oracle speeding up the overall algorithm. The efficacy
of the proposed algorithms is tested on relevant applications of sparse matrix
estimation, clustering via semidefinite relaxation, and uniform sparsest cut
problem.
