NONMONOTONE STABILIZATION METHODS FOR NONLINEAR EQUATIONS

We are concerned with defining new globalization criteria for solution methods of nonlinear equations. The current criteria used in these me thods require a sufficient decrease of a particular merit function at each iteration of the algorithm. As was observed in the field of smoot h unconstrained optimization, this descent requirement can considerabl y slow the rate of convergence of the sequence of points produced and, in some cases, can heavily deteriorate the performance of algorithms. The aim of this paper is to show that the global convergence of most methods proposed in the literature for solving systems of nonlinear eq uations can be obtained using less restrictive criteria that do not en force a monotonic decrease of the chosen merit function. In particular , we show that a general stabilization scheme, recently proposed for t he unconstrained minimization of continuously differentiable functions , can be extended to methods for the solution of nonlinear (nonsmooth) equations. This scheme includes different kinds of relaxation of the descent requirement and opens up the possibility of describing new cla sses of algorithms where the old monotone linesearch techniques are re placed with more flexible nonmonotone stabilization procedures. As in the case of smooth unconstrained optimization, this should be the basi s for defining more efficient algorithms with very good practical rate s of convergence.