Approximate Projection Methods for Decentralized Optimization with Functional Constraints

TL;DR

This paper introduces a decentralized optimization algorithm using approximate projections for problems with complex constraints, ensuring convergence and consensus among agents.

Contribution

It presents a novel decentralized method based on approximate projections and subgradient steps, suitable for non-differentiable functions and complex constraints.

Findings

Algorithm converges almost surely to a consensus solution.

Method effectively handles non-differentiable objectives and constraints.

Simulation confirms theoretical convergence results.

Abstract

We consider distributed convex optimization problems that involve a separable objective function and nontrivial functional constraints, such as Linear Matrix Inequalities (LMIs). We propose a decentralized and computationally inexpensive algorithm which is based on the concept of approximate projections. Our algorithm is one of the consensus based methods in that, at every iteration, each agent performs a consensus update of its decision variables followed by an optimization step of its local objective function and local constraints. Unlike other methods, the last step of our method is not an Euclidean projection onto the feasible set, but instead a subgradient step in the direction that minimizes the local constraint violation. We propose two different averaging schemes to mitigate the disagreements among the agents' local estimates. We show that the algorithms converge almost surely, i.e., every agent agrees on the same optimal solution, under the assumption that the objective functions and constraint functions are nondifferentiable and their subgradients are bounded. We provide simulation results on a decentralized optimal gossip averaging problem, which involves SDP constraints, to complement our theoretical results.