Abstract
This work proposes two different primal-dual splitting algorithms for solving structured monotone inclusion containing a cocoercive operator and the parallel-sum of maximally monotone operators. In particular, the parallel-sum is symmetry. The proposed primal-dual splitting algorithms are derived from two approaches: One is the preconditioned forward–backward splitting algorithm, and the other is the forward–backward–half-forward splitting algorithm. Both algorithms have a simple calculation framework. In particular, the single-valued operators are processed via explicit steps, while the set-valued operators are computed by their resolvents. Numerical experiments on constrained image denoising problems are presented to show the performance of the proposed algorithms.
Highlights
In the last decade, there has been great interest in primal-dual splitting algorithms for solving structured monotone inclusion
The reason is that many convex minimization problems arising in image processing, statistical learning, and economic management can be modelled by such monotone inclusion problems
Based on the perspective of operator splitting algorithms, these primal-dual splitting algorithms can be roughly divided into four categories: (i) Forward–backward splitting type [1,2,3,4]; (ii) Douglas-Rachford splitting type [5,6,7]; (iii) Forward–backward–forward splitting type [8,9,10,11,12]; and (iv) Projective splitting type [13,14,15,16,17]
Summary
There has been great interest in primal-dual splitting algorithms for solving structured monotone inclusion. The problem is to solve the primal inclusion m find x ∈ H such that z ∈ Ax + ∑ Li∗ ((Ki∗ ◦ Bi ◦ Ki )( Mi∗ ◦ Di ◦ Mi ))( Li x − ri ) + Cx, (3). It is easy to see that Problem 1 could be viewed as a special case of Problem 2 by letting Li = I and ri = 0, for any i = 1, · · · , m They proposed two different primal-dual algorithms for solving the primal-dual pair of monotone inclusions (3) and (4). We continue studying primal-dual splitting algorithms for solving the structured monotone inclusion (3) and (4).
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