An inertial primal‐dual fixed point algorithm for composite optimization problems

In this paper, we consider an inertial primal‐dual fixed point algorithm (IPDFP) to compute the minimizations of the sum of a non‐smooth convex function and a finite family of composite non‐smooth convex functions, each one of which is composed of a non‐smooth convex function and a bounded linear op...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences Vol. 45; no. 17; pp. 10628 - 10639
Main Authors: Wen, Meng, Tang, Yuchao, Cui, Angang, Peng, Jigen
Format: Journal Article
Language:English
Published: Freiburg Wiley Subscription Services, Inc 30.11.2022
Subjects:
Algorithms
composite optimization
Computational geometry
Convergence
Convex analysis
Convexity
Fixed points (mathematics)
Image processing
inertial algorithm
Iterative algorithms
Iterative methods
Linear operators
operator splitting
Operators (mathematics)
Optimization
Preconditioning
proximity operator
Splitting
ISSN:0170-4214, 1099-1476
Online Access:Get full text
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Summary:In this paper, we consider an inertial primal‐dual fixed point algorithm (IPDFP) to compute the minimizations of the sum of a non‐smooth convex function and a finite family of composite non‐smooth convex functions, each one of which is composed of a non‐smooth convex function and a bounded linear operator. This is a full splitting approach, in the sense that non‐smooth functions are processed individually via their proximity operators. The convergence of the IPDFP is obtained by reformulating the problem to the sum of three non‐smooth convex functions. Furthermore, we propose a preconditioning technique for the IPDFP. The key idea of the preconditioning technique is that the constant iterative parameters are updated self‐adaptively in the iteration process. What's more, we also give a simple and easy way to choose the diagonal preconditioners while the convergence of the iterative algorithms is maintained. This work brings together and notably extends several classical splitting schemes, like the primal‐dual method proposed by Chambolle and Pock, and the recent proximity algorithms of Charles et al. designed for the L1$$ {L}_1 $$/TV image denoising model. The iterative algorithm is used for solving non‐differentiable convex optimization problems arising in image processing.
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ISSN:0170-4214
1099-1476
DOI:10.1002/mma.8388