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An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

**Authors:**Dao, Minh , Phan, Hung**Date:**2021**Type:**Text , Journal article**Relation:**Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (2021), p. 1-19**Full Text:****Reviewed:****Description:**Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem. © 2021, The Author(s).

**Authors:**Dao, Minh , Phan, Hung**Date:**2021**Type:**Text , Journal article**Relation:**Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (2021), p. 1-19**Full Text:****Reviewed:****Description:**Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem. © 2021, The Author(s).

An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

**Authors:**Dao, Minh , Phan, Hung**Date:**2021**Type:**Text , Journal article**Relation:**Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (2021), p.**Full Text:****Reviewed:****Description:**Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem. © 2021, The Author(s).

**Authors:**Dao, Minh , Phan, Hung**Date:**2021**Type:**Text , Journal article**Relation:**Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (2021), p.**Full Text:****Reviewed:****Description:**Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem. © 2021, The Author(s).

Constraint reduction reformulations for projection algorithms with applications to wavelet construction

- Dao, Minh, Dizon, Neil, Hogan, Jeffrey, Tam, Matthew

**Authors:**Dao, Minh , Dizon, Neil , Hogan, Jeffrey , Tam, Matthew**Date:**2021**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 190, no. 1 (2021), p. 201-233**Full Text:****Reviewed:****Description:**We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra’s classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas–Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas–Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas–Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

**Authors:**Dao, Minh , Dizon, Neil , Hogan, Jeffrey , Tam, Matthew**Date:**2021**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 190, no. 1 (2021), p. 201-233**Full Text:****Reviewed:****Description:**We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra’s classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas–Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas–Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas–Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

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