Nonmonotone stabilization methods for nonlinear equations (Q1331105)

A new globalization criterion is defined for solution methods of nonlinear equations \(H(x) = 0\), where \(H:\mathbb{R}^ n \to\mathbb{R}^ n\) is a given function. The authors assume that there exists a locally Lipschitzian merit function \(M\) with the property that \(M(x) \geq 0\) \(\forall x \in\mathbb{R}^ n\), \(M(x) = 0\) if and only if \(H(x) = 0\). A nonmonotone stabilization algorithm is described and general conditions are given which are required for such a technique to give global convergence. These conditions are formulated in terms of a merit function, an auxiliary function, and the directions determined by a particular algorithm. The authors prove the general convergence result under these assumptions without specifying the particular merit function, the auxiliary function or the direction, but only the conditions which they must satisfy. The described conditions are so weak that almost all the merit functions and auxiliary functions in the literature satisfy the given conditions. Some examples are presented.