- Title
- Constraint reduction reformulations for projection algorithms with applications to wavelet construction
- Creator
- Dao, Minh; Dizon, Neil; Hogan, Jeffrey; Tam, Matthew
- Date
- 2021
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/178395
- Identifier
- vital:15397
- Identifier
-
https://doi.org/10.1007/s10957-021-01878-z
- Identifier
- ISBN:0022-3239 (ISSN)
- Abstract
- 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.
- Publisher
- Springer
- Relation
- Journal of Optimization Theory and Applications Vol. 190, no. 1 (2021), p. 201-233
- Rights
- All metadata describing materials held in, or linked to, the repository is freely available under a CC0 licence
- Rights
- Copyright @ 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
- Rights
- Open Access
- Subject
- 0102 Applied Mathematics; 0103 Numerical and Computational Mathematics; 0906 Electrical and Electronic EngineeringAlternating projections; Cyclic projections; Douglas–Rachford; Fixed point iterations; Wavelets
- Full Text
- Reviewed
- Funder
- The authors were partially supported by the Australian Research Council through grants DP160101537 (MND, NDD and JAH), DP190100555 (MND) and DE200100063 (MKT).
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