  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.
