{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T09:56:22Z","timestamp":1773654982931,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T00:00:00Z","timestamp":1773619200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The primal-dual hybrid gradient (PDHG) method is widely used for convex\u2013concave saddle-point problems, yet its extrapolated variants are typically asymmetric because only one side is extrapolated. We propose a symmetry-preserving refinement, E-PDHG, which performs dual-side extrapolation followed by an explicit correction step. Under standard step-size conditions, we establish global convergence for all \u03b7\u2208(\u22121,1) and derive a pointwise (non-ergodic) O(1\/t) rate for the last iterate. The method does not improve the asymptotic complexity order of PDHG; instead, it enlarges the practically stable parameter region while retaining the same per-iteration cost. Numerical experiments on image deblurring\/inpainting and additional machine learning benchmarks (logistic regression and LASSO) demonstrate improved finite-iteration stability and efficiency.<\/jats:p>","DOI":"10.3390\/axioms15030219","type":"journal-article","created":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T08:41:26Z","timestamp":1773650486000},"page":"219","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Symmetry-Preserving Extrapolated Primal-Dual Hybrid Gradient Method for Saddle-Point Problems"],"prefix":"10.3390","volume":"15","author":[{"given":"Xiayang","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, School of Cryptography, Nanjing Institute of Technology, Nanjing 211167, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenzhuo","family":"Li","sequence":"additional","affiliation":[{"name":"School of Applied Technology, Nanjing Institute of Technology, Nanjing 211167, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bowen","family":"Chang","sequence":"additional","affiliation":[{"name":"School of Applied Technology, Nanjing Institute of Technology, Nanjing 211167, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Automation, Nanjing Institute of Technology, Nanjing 211167, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-7040-5319","authenticated-orcid":false,"given":"Shiyu","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Communication and Artificial Intelligence, School of Integrated Circuits, Nanjing Institute of Technology, Nanjing 211167, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1017\/S096249291600009X","article-title":"An introduction to continuous optimization for imaging","volume":"25","author":"Chambolle","year":"2016","journal-title":"Acta Numer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2047","DOI":"10.1137\/070696143","article-title":"Efficient schemes for total variation minimization under constraints in image processing","volume":"31","author":"Weiss","year":"2009","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_3","unstructured":"Zhu, M., and Chan, T.F. (2008). An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration, UCLA. CAM Report 08-34."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1007\/s10851-010-0251-1","article-title":"A first-order primal-dual algorithms for convex problem with applications to imaging","volume":"40","author":"Chambolle","year":"2011","journal-title":"J. Math. Imaging Vis."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Pock, T., and Chambolle, A. (2011, January 6\u201313). Diagonal preconditioning for first order primal-dual algorithms in convex optimization. Proceedings of the IEEE International Conference on Computer Vision, Barcelona, Spain.","DOI":"10.1109\/ICCV.2011.6126441"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1137\/100814494","article-title":"Convergence analysis of primal-dual algorithms for a saddle-point problem: From contraction perspective","volume":"5","author":"He","year":"2012","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_7","unstructured":"Goldstein, T., Li, M., Yuan, X.M., Esser, E., and Baraniuk, R. (2013). Adaptive primal-dual hybrid gradient methods for saddle-point problems. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s10107-018-1321-1","article-title":"On the equivalence of the primal-dual hybrid gradient method and Douglas\u2013Rachford splitting","volume":"179","author":"Vandenberghe","year":"2020","journal-title":"Math. Prog."},{"key":"ref_9","unstructured":"Arrow, K.J., Hurwicz, L., and Uzawa, H. (1958). Studies in Linear and Non-Linear Programming, Stanford University Press. Stanford Mathematical Studies in the Social Science."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2526","DOI":"10.1137\/140963467","article-title":"On the convergence of primal-dual hybrid gradient algorithm","volume":"7","author":"He","year":"2014","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/s40305-015-0108-9","article-title":"PPA-like contraction methods for convex optimization: A framework using variational inequality approach","volume":"3","author":"He","year":"2015","journal-title":"J. Oper. Res. Soc. China"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1007\/s10957-012-0245-9","article-title":"A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms","volume":"158","author":"Condat","year":"2013","journal-title":"J. Optim. Theory Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1419","DOI":"10.1007\/s10898-012-9999-8","article-title":"An improved first-order primal-dual algorithm with a new correction step","volume":"57","author":"Cai","year":"2013","journal-title":"J. Glob. Optim."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/s10851-014-0523-2","article-title":"An inertial forward-backward algorithm for monotone inclusions","volume":"51","author":"Lorenz","year":"2015","journal-title":"J. Math. Imaging Vis."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1007\/s11075-015-0007-5","article-title":"An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems","volume":"71","author":"Csetnek","year":"2016","journal-title":"Numer. Algorithms"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1137\/18M1182851","article-title":"Inertial, corrected, primal-dual proximal splitting","volume":"30","author":"Valkonen","year":"2020","journal-title":"SIAM J. Optim."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s10107-015-0957-3","article-title":"On the ergodic convergence rates of a first-order primal-dual algorithm","volume":"159","author":"Chambolle","year":"2016","journal-title":"Math. Prog."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1007\/s10851-016-0692-2","article-title":"Acceleration of the PDHGM on partially strongly convex functions","volume":"59","author":"Valkonen","year":"2017","journal-title":"J. Math. Imaging Vis."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1137\/110822347","article-title":"Alternating direction method with Gaussian back substitution for separable convex programming","volume":"22","author":"He","year":"2012","journal-title":"SIAM J. Optim."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1137\/13090849X","article-title":"A strictly contractive Peaceman\u2013Rachford splitting method for convex programming","volume":"24","author":"He","year":"2014","journal-title":"SIAM J. Optim."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s10851-017-0709-5","article-title":"An algorithmic framework of generalized primal-dual hybrid gradient methods for saddle point problems","volume":"58","author":"He","year":"2017","journal-title":"J. Math. Imaging Vis."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Wu, J., and Ma, F. (2025). A primal-dual algorithm with coupled extrapolation: Bridging the Chambolle\u2013Pock and Peaceman\u2013Rachford methods. Numer. Algorithms, 1\u201339.","DOI":"10.1007\/s11075-025-02207-5"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Ma, F., Li, S., and Zhang, X. (2025). A symmetric version of the generalized Chambolle-Pock-He-Yuan method for saddle point problems. Comput. Optim. Appl., 1\u201326.","DOI":"10.1007\/s10589-025-00671-2"},{"key":"ref_24","unstructured":"Ma, F. (2020). A revisit of Chen-Teboulle\u2019s proximal-based decomposition method. arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1090\/mcom\/3822","article-title":"A rank-two relaxed parallel splitting version of the augmented Lagrangian method with step size in (0, 2) for separable convex programming","volume":"92","author":"He","year":"2023","journal-title":"Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.1007\/s11075-024-01834-8","article-title":"Preconditioned golden ratio primal-dual algorithm with linesearch","volume":"98","author":"Ma","year":"2025","journal-title":"Numer. Algorithms"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1137\/09076934X","article-title":"A general framework for a class of first order primal-dual algorithms for TV minimization","volume":"3","author":"Esser","year":"2010","journal-title":"SIAM J. Imaging Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/3\/219\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T09:00:08Z","timestamp":1773651608000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/3\/219"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,16]]},"references-count":27,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["axioms15030219"],"URL":"https:\/\/doi.org\/10.3390\/axioms15030219","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,16]]}}}