Path-following and augmented Lagrangian methods for contact problems in linear elasticity
Journal of Computational and Applied Mathematics
Computational Optimization and Applications - Special issue: Numerical analysis of optimization in partial differential equations
Computational Optimization and Applications
Update strategies for perturbed nonsmooth equations
Optimization Methods & Software
A virtual control concept for state constrained optimal control problems
Computational Optimization and Applications
Journal of Computational and Applied Mathematics
Computational Optimization and Applications
SIAM Journal on Optimization
SIAM Journal on Imaging Sciences
Computational Optimization and Applications
Semismooth Newton Methods for Optimal Control of the Wave Equation with Control Constraints
SIAM Journal on Control and Optimization
Minimal Effort Problems and Their Treatment by Semismooth Newton Methods
SIAM Journal on Control and Optimization
Multigrid second-order accurate solution of parabolic control-constrained problems
Computational Optimization and Applications
SIAM Journal on Numerical Analysis
SIAM Journal on Control and Optimization
Journal of Computational Physics
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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 [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.