Generalized forward–backward splitting with penalization for monotone inclusion problems

We introduce a generalized forward–backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximally monotone operator. We show weak ergodic co...

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Bibliographic Details
Published in:Journal of global optimization Vol. 73; no. 4; pp. 825 - 847
Main Authors: Nimana, Nimit, Petrot, Narin
Format: Journal Article
Language:English
Published: New York Springer US 15.04.2019
Springer
Springer Nature B.V
Subjects:
Computer Science
Convergence
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Operators (mathematics)
Optimization
Real Functions
Splitting
Monotone inclusion
65K05
90C25
65K10
Monotone operator
Hierarchical optimization
47H05
Forward–backward algorithm
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary:We introduce a generalized forward–backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximally monotone operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the considered monotone inclusion problem, provided that the condition corresponding to the Fitzpatrick function of the operator describing the set of the normal cone is fulfilled. Under strong monotonicity of an operator, we show strong convergence of the iterates. Furthermore, we utilize the proposed method for minimizing a large-scale hierarchical minimization problem concerning the sum of differentiable and nondifferentiable convex functions subject to the set of minima of another differentiable convex function. We illustrate the functionality of the method through numerical experiments addressing constrained elastic net and generalized Heron location problems.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-018-00730-5