  Many applications using large datasets require efficient methods for
minimizing a proximable convex function subject to satisfying a set of linear
constraints within a specified tolerance. For this task, we present a proximal
projection (PP) algorithm, which is an instance of Douglas-Rachford splitting
that directly uses projections onto the set of constraints. Formal guarantees
are presented to prove convergence of PP estimates to optimizers. Unlike many
methods that obtain feasibility asymptotically, each PP iterate is feasible.
Numerically, we show PP either matches or outperforms alternatives (e.g.
linearized Bregman, primal dual hybrid gradient, proximal augmented Lagrangian,
proximal gradient) on problems in basis pursuit, stable matrix completion,
stable principal component pursuit, and the computation of earth mover's
distances.
