Tikhonov regularized iterative methods for nonlinear problems

We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the conside...

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Published in:Optimization Vol. 73; no. 13; pp. 3787 - 3818
Main Authors: Dixit, Avinash, Sahu, D. R., Gautam, Pankaj, Som, T.
Format: Journal Article
Language:English
Published: Philadelphia Taylor & Francis 03.12.2024
Taylor & Francis LLC
Subjects:
Algorithms
Convergence
Convexity
Douglas-Rachford algorithm
Fixed points (mathematics)
Fixed points of non-expansive mappings
forward-backward algorithm
Hilbert space
Iterative methods
Operators
primal-dual algorithm
Regularization
Splitting
splitting methods
Tikhonov regularization
ISSN:0233-1934, 1029-4945
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Abstract We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of non-expansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward-backward-type algorithm and a Douglas-Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward-backward-type primal-dual algorithm and a Douglas-Rachford-type primal-dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems.
AbstractList We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of non-expansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward–backward-type algorithm and a Douglas–Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward–backward-type primal–dual algorithm and a Douglas–Rachford-type primal–dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems.
Author Sahu, D. R.
Gautam, Pankaj
Dixit, Avinash
Som, T.
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  organization: Indian Institute of Technology (BHU)
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Snippet We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve...
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SubjectTerms Algorithms
Convergence
Convexity
Douglas-Rachford algorithm
Fixed points (mathematics)
Fixed points of non-expansive mappings
forward-backward algorithm
Hilbert space
Iterative methods
Operators
primal-dual algorithm
Regularization
Splitting
splitting methods
Tikhonov regularization
Title Tikhonov regularized iterative methods for nonlinear problems
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