  We address the problem of solving convex optimization problems with many
convex constraints in a distributed setting. Our approach is based on an
extension of the alternating direction method of multipliers (ADMM) that
recently gained a lot of attention in the Big Data context. Although it has
been invented decades ago, ADMM so far can be applied only to unconstrained
problems and problems with linear equality or inequality constraints. Our
extension can handle arbitrary inequality constraints directly. It combines the
ability of ADMM to solve convex optimization problems in a distributed setting
with the ability of the Augmented Lagrangian method to solve constrained
optimization problems, and as we show, it inherits the convergence guarantees
of ADMM and the Augmented Lagrangian method.
