NDLI: Semismooth Newton Methods for Operator Equations in Function Spaces

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ulbrich, Michael
Copyright Year	2003
Abstract	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.
Starting Page	805
Ending Page	841
Page Count	37
File Format	PDF
ISSN	10526234
DOI	10.1137/S1052623400371569
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	3
Volume Number	13
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Subject Keyword	nonlinear complementarity problems superposition operators semismoothness Newton-like methods Iterative procedures optimal control problems Nonsmooth analysis generalized differentials Mathematical programming methods Newton-type methods Particular nonlinear operators superlinear convergence Complementarity and equilibrium problems and variational inequalities
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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