Semismooth Newton methods for operator equations in function spaces. (English) Zbl 1033.49039

Ulbrich, Michael

SIAM J. Optim. 13, No. 3, 805-841 (2003).

Summary: We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes nonlinear complementarity problem (NCP)-function-based reformulations of infinite-dimensional nonlinear complementarity problems and thus covers a very comprehensive class of applications. Our results generalize semismoothness and \(\alpha\)-order semismoothness from finite-dimensional spaces to a Banach space setting. For this purpose, a new infinite-dimensional generalized differential is used that is motivated by Qi’s finite-dimensional C-subdifferential [ L. Qi, Research Report AMR96/5, School of Mathematics, University of New South Wales, Australia (1996)]. We apply these semismoothness results to develop a Newton-like method for nonsmooth operator equations and prove its local q-superlinear convergence to regular solutions. If the underlying operator is \(\alpha\)-order semismooth, convergence of q-order \(1+\alpha\) is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrative examples and by applications to NCPs and a constrained optimal control problem.

MSC:

49M15	Newton-type methods
65K05	Numerical mathematical programming methods
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J52	Nonsmooth analysis
47J25	Iterative procedures involving nonlinear operators
47H30	Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)

Keywords:

Newton-like methods; semismoothness; superposition operators; generalized differentials; nonlinear complementarity problems; superlinear convergence; optimal control problems