Distributed Convex Optimization with Many Convex Constraints

TL;DR

This paper extends the ADMM algorithm to handle convex optimization problems with many arbitrary convex constraints in a distributed setting, combining advantages of ADMM and Augmented Lagrangian methods.

Contribution

It introduces a novel extension of ADMM capable of directly managing arbitrary convex inequality constraints in distributed convex optimization.

Findings

The proposed method inherits convergence guarantees of ADMM and Augmented Lagrangian methods.

It effectively handles many convex constraints in a distributed environment.

The approach is applicable to large-scale convex optimization problems.

Abstract

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.