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

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.) |