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

摘要

This paper gives a parallel computing scheme for minimizing a twice continuously differentiable function with the form ƒf(x) = ∑i = 1mƒi(xi) + ∑i = 1m∑j = 1(j > i)m ƒij(xi, xj),where x = (xT1,…,xTm)T and xi ∈ Rni, ∑mi = 1ni = 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 ƒ(x). The convergence and convergence rate are given under some conditions. The conditions for finding a global minimizer of ƒ(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.