Distributed continuous-time approximate projection protocols for shortest distance optimization problems

In this paper, we investigate a distributed shortest distance optimization problem for a multi-agent network to cooperatively minimize the sum of the quadratic distances from some convex sets, where each set is only associated with one agent. To deal with this optimization problem with projection uncertainties, we propose a distributed continuous-time dynamical protocol, where each agent can only obtain an approximate projection and communicate with its neighbors over a time-varying communication graph. First, we show that no matter how large the approximate angle is, system states are always bounded for any initial condition, and uniformly bounded with respect to all initial conditions if the inferior limit of the stepsize is greater than zero. Then, in both cases of nonempty and empty intersection of convex sets, we provide stepsize and approximate angle conditions to ensure the optimal convergence, respectively. Moreover, we also give some characterizations about the optimal solutions for the empty intersection case.