  The purpose of this paper is to introduce two new classes of accelerated
distributed proximal conjugate gradient algorithms for multi-agent constrained
optimization problems; given as minimization of a function decomposed as a sum
of M number of smooth and M number of nonsmooth functions over the common fixed
points of M number of nonlinear mappings. Exploiting the special properties of
the cost component function of the objective function and the nonlinear mapping
of the constraint problem of each agent, a new inertial accelerated incremental
and parallel computing distributed algorithms will be presented based on the
combinations of computations of proximal, conjugate gradient and Halpern
methods. Some numerical experiments and comparisons are given to illustrate our
results.
