We address the problem of solving convex optimization problems with many non-linear constraints in a distributed setting. Our approach is based on an extension of the alternating direction method of multipliers (ADMM). 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 directly handle arbitrary inequality constraints. 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 both ADMM and the Augmented Lagrangian method.
For example, REF studied the direct extension of ADMM to two-block convex problems with coupled non-linear but separable constraints.
Distributed Convex Optimization with Many Non-Linear Constraints