Abstract. For a version of Newton's method applied to a generalized equation with a parameter, we extend the paradigm of the Lyusternik-Graves theorem to the framework of a mapping acting from the pair "parameter-starting point" to the set of corresponding convergent Newton sequences. Under ample parameterization, metric regularity of the mapping associated with convergent Newton sequences becomes equivalent to the metric regularity of the mapping associated with the generalized equation. We also discuss an inexact Newton method and present an application to discretized optimal control.
International audienceWe investigate the existence of several kinds of prederivatives for set-valued mappings enjoying convex properties. Then, we established both necessary and sufficient optimality conditions, involving such prederivatives, for set optimization problems
We consider some metric regularity properties of order q for set-valued mappings and we establish several characterizations of these concepts in terms of Hölder-like properties of the inverses of the mappings considered. In addition, we show that even if these properties are weaker than the classical notions of regularity for set-valued maps, they allow us to solve variational inclusions under mild assumptions.
We study the Tikhonov regularization for perturbed inclusions of the form T (x) y * where T is a set-valued mapping defined on a Banach space that enjoys metric regularity properties and y * is an element near 0. We investigate the case when T is metrically regular and strongly regular and we show the existence of both a solution x * to the perturbed inclusion and a Tikhonov sequence which converges to x * . Finally, we show that the Tikhonov sequences associated to the perturbed problem inherit the regularity properties of the inverse of T .
a b s t r a c tWe employ recent developments of generalized differentiation concepts for set-valued mappings and present a Newton-like iteration for solving generalized equations of the formboth of them being smooth mappings acting between two general Banach spaces X and Y . The Newton iteration we propose is constructed on the basis of a linearization of both f and F ; we prove that, under suitable assumptions on the ''derivatives'' of f and F , it converges Q-linearly to a solution to the generalized equation in question. When we strengthen our assumptions, we obtain the Q-quadratic convergence of the method.
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