
Journal of Convex Analysis 29 (2022), No. 1, 183204 Copyright Heldermann Verlag 2022 A Hybrid Semismooth QuasiNewton Method for Structured Nonsmooth Operator Equations in Banach Spaces Florian Mannel University of Graz, Heinrichstr. 36, 8010 Graz, Austria florian.mannel@unigraz.at Armin Rund University of Graz, Heinrichstr. 36, 8010 Graz, Austria armin.rund@unigraz.at We present an algorithm for the solution of structured nonsmooth operator equations in Banach spaces. Specifically, we seek roots of mappings that involve the composition of a smooth outer and a semismooth inner map. To exploit this structure we propose a hybrid approach in which the semismooth part is linearized in the same way as in semismooth Newton methods while the smooth part is handled by a Broydenlike method. The resulting algorithm is a semismooth Newtontype method that does not require the evaluation of the derivative of the smooth part. We prove local qlinear and qsuperlinear convergence results for the hybrid algorithm. In particular, this is the first work that establishes superlinear convergence of a semismooth quasiNewton method in an infinitedimensional setting. The convergence results also extend known finitedimensional ones in that the structure of the equation and the algorithm under consideration are more general than those available in the literature. In addition, it is shown that qlinear convergence of the iterates and compactness of the initial operator discrepancy of the smooth part implies qsuperlinear convergence without the assumption that the initial operator discrepancy is small in norm, which is a new type of result for semismooth quasiNewton methods. The convergence theory is developed under mild assumptions, which yields extensions of available results for semismooth quasiNewton methods as well as for Broydenlike methods. The benefit of the method in practical applications is addressed in a complementary paper. There, we show on problems from optimal control that the assumptions for qsuperlinear convergence are satisfied and that the hybrid approach leads to highly competitive numerical schemes that have substantially lower runtimes than stateoftheart semismooth Newton methods. Keywords: Semismooth Newtontype methods, Broydenlike method, quasiNewton methods, superlinear convergence, nonsmooth operator equations. MSC: 47J25, 47N10, 49J27, 49J52, 49M15, 49M27, 65J15, 90C30, 90C48, 90C53, 90C56. [ Fulltextpdf (180 KB)] for subscribers only. 