Nonmonotone Trust Region Methods For Nonsmooth Equations With Box Constraints

Trust region method is a class of highly effective numerical methods for nonlinearoptimization, which is very reliable and robust and has very strong convergent proper-ties. Since the idea of nonmonotone methods abandons the restriction of the descentproperty of the value of the objective function, which allows the sequence of iterates tofollow the bottom of curved narrow valleys (a common occurrence in difficult nonlinearproblems) much more loosely, which hopefully results in longer and more efficient steps.Ulbrich et al present trust region methods with the stronger nonmonotone structure.Also, we present trust region methods for nonsmooth equations with box constrains,which uses the standard nonmonotone structure. The radius of trust region is different,which is relative to the message of the current iterate point. We prove the strong con-vergence of the methods, that is, if the methods do not terminate in finite iterations,then every limit point of the sequence generated by the methods is a stationary pointof the primal problem.At last, we do some numerical experiments for nonmonotone trust region methodwith Matlab language. The numerical results are analyzed. The algorithm is veryeffective. It is worth notice that the nonmonotone methods may get the global solutionfor some problems.