Double smoothing for time-varying distributed multiuser optimization

Constrained optimization problems that couple different cooperating users sharing the same communication network are often referred to as multiuser optimization programs. We are interested in convex discrete-time time-varying multiuser optimization, where the problem to be solved changes at each time step. We study a distributed algorithm to generate a sequence of approximate optimizers of these problems. The algorithm requires only one round of communication among neighboring users between subsequent time steps and, under mild assumptions, converges linearly to a bounded error floor whose size is dependent on the variability of the optimization problem in time. To develop the algorithm we employ a double regularization both in the primal and in the dual space. This increases the convergence rate and helps us in the convergence proof. Numerical results support the theoretical findings.