Distributed Optimization with Projection-free Dynamics

We consider continuous-time dynamics for distributed optimization with set constraints in the note. 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 with solving an alternative linear optimization instead of a quadratic one. To make the algorithm implementable over weight-balanced digraphs, we design one 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.