Distributed Optimization With Projection-Free Dynamics: A Frank-Wolfe Perspective

We consider solving distributed constrained optimization in this article. To avoid projection operations due to constraints in the scenario with large-scale variable dimensions, we propose distributed projection-free dynamics by employing the Frank-Wolfe method, also known as the conditional gradient. Technically, we find a feasible descent direction by solving an alternative linear suboptimization. To make the approach available over multiagent networks with weight-balanced digraphs, we design dynamics to simultaneously achieve both the consensus of local decision variables and the global gradient tracking of auxiliary variables. Then, we present the rigorous convergence analysis of the continuous-time dynamical systems. Also, we derive its discrete-time scheme with an accordingly proved convergence rate of $O(1/k)$ . Furthermore, to clarify the advantage of our proposed distributed projection-free dynamics, we make detailed discussions and comparisons with both existing distributed projection-based dynamics and other distributed Frank-Wolfe algorithms.