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In this paper, we investigate the distributed shortest distance optimizationproblem for a multi-agent network to cooperatively minimize the sum of thequadratic distances from some convex sets, where each set is only associatedwith one agent. To deal with the optimization problem with projectionuncertainties, we propose a distributed continuous-time dynamical protocolbased on a new concept of approximate projection. Here each agent can onlyobtain an approximate projection point on the boundary of its convex set, andcommunicate with its neighbors over a time-varying communication graph. First,we show that no matter how large the approximate angle is, the system statesare always bounded for any initial condition, and uniformly bounded withrespect to all initial conditions if the inferior limit of the stepsize isgreater than zero. Then, in the two cases, nonempty intersection and emptyintersection of convex sets, we provide stepsize and approximate angleconditions to ensure the optimal convergence, respectively. Moreover, we givesome characterizations about the optimal solutions for the empty intersectioncase and also present the convergence error between agents' estimates and theoptimal point in the case of constant stepsizes and approximate angles.

Distributed Continuous-time Approximate Projection Protocols for Shortest Distance Optimization Problems