{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:36:14Z","timestamp":1760236574458,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,13]],"date-time":"2021-12-13T00:00:00Z","timestamp":1639353600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Natural Science Foundations of China","award":["12061045, 11661056"],"award-info":[{"award-number":["12061045, 11661056"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2013backward splitting algorithm, and the other is the forward\u2013backward\u2013half-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.<\/jats:p>","DOI":"10.3390\/sym13122415","type":"journal-article","created":{"date-parts":[[2021,12,14]],"date-time":"2021-12-14T01:22:05Z","timestamp":1639444925000},"page":"2415","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Primal-Dual Splitting Algorithms for Solving Structured Monotone Inclusion with Applications"],"prefix":"10.3390","volume":"13","author":[{"given":"Jinjian","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Nanchang University, Nanchang 330031, China"}]},{"given":"Xingyu","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Nanchang University, Nanchang 330031, China"}]},{"given":"Yuchao","family":"Tang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Nanchang University, Nanchang 330031, China"}]},{"given":"Qiaoli","family":"Dong","sequence":"additional","affiliation":[{"name":"Tianjin Key Laboratory for Advanced Signal Processing and College of Science, Civil Aviation University of China, Tianjin 300300, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1007\/s10444-011-9254-8","article-title":"A splitting algorithm for dual monotone inclusions involving cocoercive operators","volume":"38","year":"2013","journal-title":"Adv. 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