Inexact Newton Methods and Mixed Nonlinear Complementary Problems

In this paper we present the results obtained in the solution of sparse and large systems of nonlinear equations by Inexact Newton-like methods [6]. The linearized systems are solved with two preconditioners particularly suited for parallel computation. We report the results for the solution of some nonlinear problems on the CRAY T3E under the MPI environment. Our methods may be used to solve more general problems. Due to the presence of a logarithmic penalty, the interior point solution [10] of a nonlinear mixed complementary problem [7] can indeed be viewed as a variant of an Inexact Newton method applied to a particular system of nonlinear equations. We have applied this inexact interior point algorithm for the solution of some nonlinear complementary problems. We provide numerical results in both sequential and parallel implementations. 1 The Inexact Newton-Cimmino method Consider the system on nonlinear equations G(x) = 0 G = (g1, ..., gn) T (1) where G : R → R is a nonlinear C function, and its Jacobian matrix J(x). For solving (1) we use an iterative procedure which combines a Newton and a Quasi-Newton method with a row-projection (or row-action) linear solver of Cimmino type [11], particularly suited for parallel computation. Here below, referring to block Cimmino method, we give the general lines of this procedure. Let As = b be the linearized system to be solved. Let us partition A into p row-blocks: Ai, i = 1, . . . , p, i.e. A T = [A1, A2, . . . , Ap] and partition the vector b conformally. Then the original system is premultiplied (preconditioning) by Hp = [A + 1 , . . . , A + i , . . . , A + p ] (2) where A+i = A T i (AiA T i ) −1 is the Moore-Penrose pseudo inverse of Ai. We obtain the equivalent system HpAs = Hpb, (P1 + · · ·+ Pp)s = p