Abstract
Superlinear convergence of the Newton method for nonsmooth equations requires a "semismoothness" assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ-order semismooth, where γ is a positive parameter. As an application of this new estimate, we prove that the error at the kth step of the Newton method behaves like O(2-(1+γ)k). © 2007 Springer-Verlag.
Original language | English |
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Pages (from-to) | 5-19 |
Journal | Mathematical Programming |
Volume | 117 |
Issue number | 1-2 |
DOIs | |
Publication status | Published - 1 Mar 2009 |
Keywords
- Nonsmooth Newton method
- O-minimal structure
- Semi-algebraic function
- Semismoothness
- Structured optimization problem
- Superlinear convergence