Journal of Convex Analysis 29 (2022), No. 1, 183--204
Copyright Heldermann Verlag 2022
A Hybrid Semismooth Quasi-Newton Method for Structured Nonsmooth Operator Equations in Banach Spaces
Florian Mannel
University of Graz, Heinrichstr. 36, 8010 Graz, Austria
florian.mannel@uni-graz.at
Armin Rund
University of Graz, Heinrichstr. 36, 8010 Graz, Austria
armin.rund@uni-graz.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 Broyden-like method. The resulting algorithm is a semismooth Newton-type method that does not require the evaluation of the derivative of the smooth part.
We prove local q-linear and q-superlinear convergence results for the hybrid algorithm. In particular, this is the first work that establishes superlinear convergence of a semismooth quasi-Newton method in an infinite-dimensional setting. The convergence results also extend known finite-dimensional 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 q-linear convergence of the iterates and compactness of the initial operator discrepancy of the smooth part implies q-superlinear convergence without the assumption that the initial operator discrepancy is small in norm, which is a new type of result for semismooth quasi-Newton methods. The convergence theory is developed under mild assumptions, which yields extensions of available results for semismooth quasi-Newton methods as well as for Broyden-like 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 q-superlinear convergence are satisfied and that the hybrid approach leads to highly competitive numerical schemes that have substantially lower runtimes than state-of-the-art semismooth Newton methods.
Keywords: Semismooth Newton-type methods, Broyden-like method, quasi-Newton methods, superlinear convergence, nonsmooth operator equations.
MSC: 47J25, 47N10, 49J27, 49J52, 49M15, 49M27, 65J15, 90C30, 90C48, 90C53, 90C56.
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