A proximal-Newton method for unconstrained convex optimization in Hilbert spaces

Optimization ◽  
2017 ◽  
Vol 67 (1) ◽  
pp. 67-82 ◽  
Author(s):  
M. Marques Alves ◽  
Benar F. Svaiter
Keyword(s):  
Convex Optimization ◽  
Newton Method ◽  
Hilbert Spaces

scholarly journals A LEPSKIJ-TYPE STOPPING-RULE FOR SIMPLIFIED ITERATIVELY REGULARIZED GAUSS-NEWTON METHOD

2017 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Newton Method ◽  
Hilbert Spaces ◽  
Potential Problem ◽  
Initial Approximation ◽  
Stopping Rule ◽  
A Posteriori ◽  
Iterative Regularization ◽  
Ill Posed ◽  
Derivatives Of ◽  
Unknown Solution

Iterative regularization methods for nonlinear ill-posed equations of the form $ F(a)= y$, where $ F: D(F) \subset X \to Y$ is an operator between Hilbert spaces $ X $ and $ Y$, usually involve calculation of the Fr\'{e}chet derivatives of $ F$ at each iterate and at the unknown solution $ a^\sharp$. A modified form of the generalized Gauss-Newton method which requires the Fr\'{e}chet derivative of $F$ only at an initial approximation $ a_0$ of the solution $ a^\sharp$ as studied by Mahale and Nair \cite{MaNa:2k9}. This work studied an {\it a posteriori} stopping rule of Lepskij-type of the method. A numerical experiment from inverse source potential problem is demonstrated.

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