Abstract
It is well-recognized that in the presence of singular (and in particular nonisolated) solutions of unconstrained or constrained smooth nonlinear equations, the existence of critical solutions has a crucial impact on the behavior of various Newton-type methods. On the one hand, it has been demonstrated that such solutions turn out to be attractors for sequences generated by these methods, for wide domains of starting points, and with a linear convergence rate estimate. On the other hand, the pattern of convergence to such solutions is quite special, and allows for a sharp characterization which serves, in particular, as a basis for some known acceleration techniques, and for the proof of an asymptotic acceptance of the unit stepsize. The latter is an essential property for the success of these techniques when combined with a linesearch strategy for globalization of convergence. This paper aims at extensions of these results to piecewise smooth equations, with applications to corresponding reformulations of nonlinear complementarity problems.
Highlights
In the recent publications [10, 11, 17], it has been demonstrated that for smooth nonlinear equations with singular solutions, the local behavior of various Newton-type methods is strongly affected by the existence of critical solutions, as defined in [18]
[17] extends the results in [14] from the basic Newton method to perturbed versions, establishing their local convergence to a solution satisfying a certain 2-regularity property, from wide domains of starting points. This framework covers a large range of Newton-type methods, including those supplied with stabilization mechanisms, and developed especially for tackling the case of nonisolated solutions, like the Levenberg–Marquardt method [22, 23], the LP-Newton method [5], and stabilized sequential quadratic programming for optimization [7, 16, 19, 25]
Under the mentioned 2-regularity property, the convergence rate of the methods in the framework is linear with a common ratio of 1/2, and cannot be any faster, whereas the 2-regularity requirement can only hold at those singular solutions that are critical
Summary
In the recent publications [10, 11, 17], it has been demonstrated that for smooth nonlinear equations with singular (and possibly nonisolated) solutions, the local behavior of various Newton-type methods is strongly affected by the existence of critical solutions, as defined in [18]. The results in this work strongly rely on those in [10, 11, 17], and are obtained by combining techniques from these references with specificities of piecewise smooth structures These combinations involve some rather subtle ingredients, like choosing a direction with appropriate collection of active smooth selections associated to it in Theorems 1 and 2, the role of condition (32) for globalization issues, separation of constraints into two parts in Theorem 3, etc. All these ingredients are crucial to cover a much wider territory in [10, 11, 17], which is demonstrated by rather nontrivial direct (i.e., not requiring any decompositions) applications to complementarity problems in Sect. RU(u) stands for the radial cone to a set U at u ∈ U , i.e., the set of directions v ∈ Rp such that u + tv ∈ U for all t > 0 small enough
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