Distributed constrained optimization and consensus in uncertain networks via proximal minimization

TL;DR

This paper introduces a unified proximal minimization framework for distributed convex optimization over dynamic networks with constraints and uncertainty, simplifying analysis and ensuring probabilistic feasibility guarantees.

Contribution

It develops a novel distributed algorithm that handles time-varying networks, constraints, and uncertainty simultaneously, with convergence and probabilistic feasibility guarantees.

Findings

Algorithm converges to a centralized optimizer.

Distributed implementation of scenario-based uncertainty handling.

Numerical example demonstrates effectiveness in regression with regularization.

Abstract

We provide a unifying framework for distributed convex optimization over time-varying networks, in the presence of constraints and uncertainty, features that are typically treated separately in the literature. We adopt a proximal minimization perspective and show that this set-up allows us to bypass the difficulties of existing algorithms while simplifying the underlying mathematical analysis. We develop an iterative algorithm and show convergence of the resulting scheme to some optimizer of the centralized problem. To deal with the case where the agents' constraint sets are affected by a possibly common uncertainty vector, we follow a scenario-based methodology and offer probabilistic guarantees regarding the feasibility properties of the resulting solution. To this end, we provide a distributed implementation of the scenario approach, allowing agents to use a different set of uncertainty scenarios in their local optimization programs. The efficacy of our algorithm is demonstrated by means of a numerical example related to a regression problem subject to regularization.