{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T09:40:27Z","timestamp":1777542027927,"version":"3.51.4"},"reference-count":27,"publisher":"American Mathematical Society (AMS)","issue":"345","license":[{"start":{"date-parts":[[2024,5,17]],"date-time":"2024-05-17T00:00:00Z","timestamp":1715904000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We propose an adaptive way to choose the anchoring parameters for the Halpern iteration to find a fixed point of a nonexpansive mapping in a real Hilbert space. We prove strong convergence of this adaptive Halpern iteration and obtain the rate of asymptotic regularity at least\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis 1 slash k right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(1\/k)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the number of iterations. Numerical experiments are also provided to show advantages and outperformance of our adaptive Halpern algorithm over the standard Halpern algorithm.\n                  <\/p>","DOI":"10.1090\/mcom\/3851","type":"journal-article","created":{"date-parts":[[2023,4,5]],"date-time":"2023-04-05T09:15:37Z","timestamp":1680686137000},"page":"327-345","source":"Crossref","is-referenced-by-count":9,"title":["Convergence analysis of the Halpern iteration with adaptive anchoring parameters"],"prefix":"10.1090","volume":"93","author":[{"given":"Songnian","family":"He","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong-Kun","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiao-Li","family":"Dong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Na","family":"Mei","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2023,5,17]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1080\/10556788.2018.1457151","article-title":"Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces","volume":"34","author":"Bo\u0163, Radu Ioan","year":"2019","journal-title":"Optim. Methods Softw.","ISSN":"https:\/\/id.crossref.org\/issn\/1055-6788","issn-type":"print"},{"key":"2","unstructured":"H. Cheval, U. Kohlenbach, and L. Leu\u015ftean, On modifined Halpern and Tikhonov-Mann iterations,  arXiv:2203.11003v3[math.OC], 11 Apr 2022."},{"key":"3","doi-asserted-by":"publisher","first-page":"957","DOI":"10.1090\/S0002-9904-1967-11864-0","article-title":"Fixed points of nonexpanding maps","volume":"73","author":"Halpern, Benjamin","year":"1967","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"key":"4","unstructured":"J. Diakonikolas, Halpern iteration for near-optimal and parameter-free monotone inclusion and strong solutions to variational inequalities,  arXiv:2002.08872v3. Proceedings of Thirty Third Conference on Learning Theory, PMLR 125, pp. 1428\u20131451, 2020."},{"key":"5","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511526152","volume-title":"Topics in metric fixed point theory","volume":"28","author":"Goebel, Kazimierz","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0521382890"},{"key":"6","series-title":"Monographs and Textbooks in Pure and Applied Mathematics","isbn-type":"print","volume-title":"Uniform convexity, hyperbolic geometry, and nonexpansive mappings","volume":"83","author":"Goebel, Kazimierz","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0824772237"},{"issue":"9","key":"7","doi-asserted-by":"publisher","first-page":"1959","DOI":"10.1080\/02331934.2020.1767101","article-title":"On the optimal relaxation parameters of Krasnosel\u2019ski\u012d-Mann iteration","volume":"70","author":"He, Songnian","year":"2021","journal-title":"Optimization","ISSN":"https:\/\/id.crossref.org\/issn\/0233-1934","issn-type":"print"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"1171","DOI":"10.1007\/s11075-018-00650-1","article-title":"Optimal parameter selections for a general Halpern iteration","volume":"82","author":"He, Songnian","year":"2019","journal-title":"Numer. 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