  This paper considers a networked system with a finite number of users and
supposes that each user tries to minimize its own private objective function
over its own private constraint set. It is assumed that each user's constraint
set can be expressed as a fixed point set of a certain quasi-nonexpansive
mapping. This enables us to consider the case in which the projection onto the
constraint set cannot be computed efficiently. This paper proposes two methods
for solving the problem of minimizing the sum of their nondifferentiable,
convex objective functions over the intersection of their fixed point sets of
quasi-nonexpansive mappings in a real Hilbert space. One method is a parallel
subgradient method that can be implemented under the assumption that each user
can communicate with other users. The other is an incremental subgradient
method that can be implemented under the assumption that each user can
communicate with its neighbors. Investigation of the two methods' convergence
properties for a constant step size reveals that, with a small constant step
size, they approximate a solution to the problem. Consideration of the case in
which the step-size sequence is diminishing demonstrates that the sequence
generated by each of the two methods strongly converges to the solution to the
problem under certain assumptions. Convergence rate analysis of the two methods
under certain situations is provided to illustrate the two methods' efficiency.
This paper also discusses nonsmooth convex optimization over sublevel sets of
convex functions and provides numerical comparisons that demonstrate the
effectiveness of the proposed methods.
