Decentralized Constraint-Coupled Optimization with Inexact Oracle

TL;DR

This paper introduces iDDGT, an inexact decentralized dual gradient tracking method that improves computational efficiency and converges linearly over directed graphs without strict matrix conditions.

Contribution

The paper presents a novel inexact dual gradient approach for decentralized optimization with coupled constraints, extending convergence guarantees to directed graphs.

Findings

iDDGT achieves higher computational efficiency than existing methods.

It guarantees linear convergence over directed graphs without full row rank constraint.

The method is applicable to a broader class of problems with coupled constraints.

Abstract

We propose an inexact decentralized dual gradient tracking method (iDDGT) for decentralized optimization problems with a globally coupled equality constraint. Unlike existing algorithms that rely on either the exact dual gradient or an inexact one obtained through single-step gradient descent, iDDGT introduces a new approach: utilizing an inexact dual gradient with controllable levels of inexactness. Numerical experiments demonstrate that iDDGT achieves significantly higher computational efficiency compared to state-of-the-art methods. Furthermore, it is proved that iDDGT can achieve linear convergence over directed graphs without imposing any conditions on the constraint matrix. This expands its applicability beyond existing algorithms that require the constraint matrix to have full row rank and undirected graphs for achieving linear convergence.