Distributed Optimization with Projection-free Dynamics

TL;DR

This paper introduces a continuous-time distributed optimization method that avoids projections by using the Frank-Wolfe algorithm, enabling efficient solutions over directed graphs with proven convergence.

Contribution

It proposes a novel projection-free distributed dynamics based on Frank-Wolfe, suitable for weight-balanced digraphs, with convergence proofs and discrete-time implementation.

Findings

Convergence to optimal solutions is proven for the proposed dynamics.

Numerical comparisons show improved efficiency over projection-based methods.

A discrete-time scheme with proven convergence rate is derived.

Abstract

We consider continuous-time dynamics for distributed optimization with set constraints in the paper. To handle the computational complexity of projection-based dynamics due to solving a general quadratic optimization subproblem with projection, we propose a distributed projection-free dynamics by employing the Frank-Wolfe method, also known as the conditional gradient algorithm. The process searches a feasible descent direction by solving an alternative linear optimization instead of a quadratic one. To make the approach applicable over weight-balanced digraphs, we design a dynamics for the consensus of local decision variables and another dynamics of auxiliary variables to track the global gradient. Then we prove the convergence of the dynamical systems to the optimal solution, and provide detailed numerical comparisons with both projection-based dynamics and other distributed projection-free algorithms. Also, we derive the distributed discrete-time scheme following the instructive ideas of the proposed dynamics and provide its accordingly convergence rate.