A parallel computing scheme for minimizing a class of large scale functions

This paper gives a parallel computing scheme for minimizing a twice continuously differentiable function with the form

where x = (x^T₁,…,x^T_m)^T and x_i R^n_i, ∑^m_{i = 1}n_i = n, and n a very big number. It is proved that we may use m parallel processors and an iterative procedure to find a minimizer of ?(x). The convergence and convergence rate are given under some conditions. The conditions for finding a global minimizer of ?(x by using this scheme are given, too. A similar scheme can also be used parallelly to solve a large scale system of nonlinear equations in the similar way. A more general case is also investigated.