Document Zbl 1543.47129

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The <strong>and -operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell" role="cell">The <strong>or</strong>-operator <strong>|</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3AEur%2A+J%2A+Mat%2A+Soc%2A+cc%3A14">so:Eur* J* Mat* Soc* cc:14</a></span> <span class="search-explanation" role="cell" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%27%27Partial+diff%2A+eq%2A%27%27+%21elliptic">''Partial diff* eq*'' !elliptic</a></span> <span class="search-explanation" role="cell" role="cell">The <strong>not</strong> -operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to <u>b</u>ooks; alternatively: <u>j</u> for <strong><u>j</u>ournal articles</strong>, <u>a</u> for <strong>book <u>a</u>rticles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000+-+2015+cc%3A%2894A+%7C+11T%29">py: 2000 - 2015 cc:(94A | 11T)</a></span> <span class="search-explanation" role="cell" role="cell">Number <strong>ranges</strong> when searching for <strong><u>p</u>ublication <u>y</u>ear</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href='https://en.wikipedia.org/wiki/ISO_639-1'>ISO 639 - 1</a> (opens in new tab) language codes can also be used.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=st%3A+c+r+s">st: c r s</a></span> <span class="search-explanation" role="cell" role="cell">Find documents that are <strong><u>c</u>ited</strong>, have <strong><u>r</u>eferences</strong> and are from a <strong><u>s</u>ingle author</strong>.</span> </div> </div> <p style="margin-bottom:0.5cm;"></p> <h2>Fields</h2> <div id="help-fields" role="table"> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ab</strong> </span> <span class="search-explanation" role="cell" role="cell"> Text from the summary or review (for phrases use “...”) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>an</strong> </span> <span class="search-explanation" role="cell" role="cell"> zbMATH ID, i.e.: preliminary ID, Zbl number, JFM number, ERAM number </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>any</strong> </span> <span class="search-explanation" role="cell" role="cell"> Includes ab, au, cc, en, rv, so, ti, ut </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>arxiv</strong> </span> <span class="search-explanation" role="cell" role="cell"> arXiv preprint number </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>au</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name(s) of the contributor(s) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>br</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of a person with biographic references (to find documents about the life or work) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>cc</strong> </span> <span class="search-explanation" role="cell" role="cell"> Code from the Mathematics Subject Classification (prefix with <strong>*</strong> to search only primary MSC) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ci</strong> </span> <span class="search-explanation" role="cell" role="cell"> zbMATH ID of a document cited in summary or review </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>db</strong> </span> <span class="search-explanation" role="cell" role="cell"> Database: documents in Zentralblatt für Mathematik/zbMATH Open (<em>db:Zbl</em>), Jahrbuch über die Fortschritte der Mathematik (<em>db:JFM</em>), Crelle's Journal (<em>db:eram</em>), arXiv (<em>db:arxiv</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>dt</strong> </span> <span class="search-explanation" role="cell" role="cell"> Type of the document: journal article (<em>dt:j</em>), collection article (<em>dt:a</em>), book (<em>dt:b</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>doi</strong> </span> <span class="search-explanation" role="cell" role="cell"> Digital Object Identifier (DOI) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ed</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of the editor of a book or special issue </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>en</strong> </span> <span class="search-explanation" role="cell" role="cell"> External document ID: DOI, arXiv ID, ISBN, and others </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>in</strong> </span> <span class="search-explanation" role="cell" role="cell"> zbMATH ID of the corresponding issue </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>la</strong> </span> <span class="search-explanation" role="cell" role="cell"> Language (use name, e.g., <em>la:French</em>, or <a href='https://en.wikipedia.org/wiki/ISO_639-1'>ISO 639-1</a>, e.g., <em>la:FR)</em> </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>li</strong> </span> <span class="search-explanation" role="cell" role="cell"> External link (URL) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>na</strong> </span> <span class="search-explanation" role="cell" role="cell"> Number of authors of the document in question. Interval search with “-” </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>pt</strong> </span> <span class="search-explanation" role="cell" role="cell"> Reviewing state: Reviewed (<em>pt:r</em>), Title Only (<em>pt:t</em>), Pending (<em>pt:p</em>), Scanned Review (<em>pt:s</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>pu</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of the publisher </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>py</strong> </span> <span class="search-explanation" role="cell" role="cell"> Year of publication. Interval search with “-” </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>rft</strong> </span> <span class="search-explanation" role="cell" role="cell"> Text from the references of a document (for phrases use “...”) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>rn</strong> </span> <span class="search-explanation" role="cell" role="cell"> Reviewer ID </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>rv</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name or ID of the reviewer </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>se</strong> </span> <span class="search-explanation" role="cell" role="cell"> Serial ID </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>si</strong> </span> <span class="search-explanation" role="cell" role="cell"> swMATH ID of software referred to in a document </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>so</strong> </span> <span class="search-explanation" role="cell" role="cell"> Bibliographical source, e.g., serial title, volume/issue number, page range, year of publication, ISBN, etc. </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>st</strong> </span> <span class="search-explanation" role="cell" role="cell"> State: is cited (<em>st:c</em>), has references (<em>st:r</em>), has single author (<em>st:s</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>sw</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of software referred to in a document </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ti</strong> </span> <span class="search-explanation" role="cell" role="cell"> Title of the document </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ut</strong> </span> <span class="search-explanation" role="cell" role="cell"> Keywords </span> </div> </div> <p style="margin-bottom:0.5cm;"></p> <h2>Operators</h2> <div id="operators-fields" role="table"> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>a & b</strong></span> <span class="search-explanation" role="cell" role="cell">logical and</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>a | b</strong></span> <span class="search-explanation" role="cell" role="cell">logical or</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>!ab</strong></span> <span class="search-explanation" role="cell" role="cell">logical not</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>abc*</strong></span> <span class="search-explanation" role="cell" role="cell">right wildcard</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>ab c</strong></span> <span class="search-explanation" role="cell" role="cell">phrase</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>(ab c)</strong></span> <span class="search-explanation" role="cell" role="cell">parentheses</span> </div> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/ryu.ernest-k" title="Author Profile">Ryu, Ernest K.</a><a class="btn btn-default btn-xs matrix" type="button" href="https://donald.zentralblatt-math.org/lethe/recruit/?first_name=Ernest+K.&last_name=Ryu&document_id=7550212"> <span class="text-danger">Recruit</span> </a>; <a href="/authors/hannah.robert" title="Author Profile">Hannah, Robert</a><a class="btn btn-default btn-xs matrix" type="button" href="https://donald.zentralblatt-math.org/lethe/recruit/?first_name=Robert&last_name=Hannah&document_id=7550212"> <span class="text-danger">Recruit</span> </a>; <a href="/authors/yin.wotao" title="Author Profile">Yin, Wotao</a><a class="btn btn-default btn-xs matrix" type="button" href="https://donald.zentralblatt-math.org/lethe/recruit/?first_name=Wotao&last_name=Yin&document_id=7550212"> <span class="text-danger">Recruit</span> </a></div> <h2 class="title"> <strong>Scaled relative graphs: nonexpansive operators via 2D Euclidean geometry.</strong> <i>(English)</i> <a class="label nowrap" href="/1543.47129">Zbl 1543.47129</a> </h2> <div class="source"> <a href="/serials/1746" title="Journal Profile">Math. Program.</a> <a href="/?q=in%3A483696" title="Articles in this Issue">194, No. 1-2 (A), 569-619 (2022)</a>. </div> <div class="abstract"><div class="pre">Summary: Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph. The SRG provides a correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of convergence.</div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7550212">Cited in <strong>4</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47J26" title="MSC2020">47J26</a> </td> <td class="space"> Fixed-point iterations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47H05" title="MSC2020">47H05</a> </td> <td class="space"> Monotone operators and generalizations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47H09" title="MSC2020">47H09</a> </td> <td class="space"> Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A51M04" title="MSC2020">51M04</a> </td> <td class="space"> Elementary problems in Euclidean geometries </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C25" title="MSC2020">90C25</a> </td> <td class="space"> Convex programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49M27" title="MSC2020">49M27</a> </td> <td class="space"> Decomposition methods </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afixed-point+iteration">fixed-point iteration</a>; <a href="/?q=ut%3AEuclidean+geometry">Euclidean geometry</a>; <a href="/?q=ut%3Ainversive+geometry">inversive geometry</a>; <a href="/?q=ut%3Acontraction+mapping">contraction mapping</a>; <a href="/?q=ut%3ADouglas-Rachford+splitting">Douglas-Rachford splitting</a>; <a href="/?q=ut%3Ametric+subregularity">metric subregularity</a>; <a href="/?q=ut%3Amonotone+operator">monotone operator</a></div> <div class="software"> <h3>Software:</h3><a href="/software/7020">Eigtool</a>; <a href="/software/20864">PESTO</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1543.47129" data-ciurl="/ci/07550212" data-biburl="/bibtex/07550212.bib" data-amsurl="/amsrefs/07550212.bib" data-xmlurl="/xml/07550212.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07550212.pdf" title="Zbl 1543.47129 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s10107-021-01639-w" aria-label="DOI for “Scaled relative graphs: nonexpansive operators via 2D Euclidean geometry”" title="10.1007/s10107-021-01639-w">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1902.09788" title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ablowitz, MJ; Fokas, AS, Complex Variables: Introduction and Applications (2003), Cambridge: Cambridge University Press, Cambridge · <a href="/1088.30001" class="nowrap">Zbl 1088.30001</a></td> </tr><tr> <td>[2]</td> <td class="space">Abramowitz, M.; Stegun, IA, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1964), New York: Dover, New York · <a href="/0171.38503" class="nowrap">Zbl 0171.38503</a></td> </tr><tr> <td>[3]</td> <td class="space">Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3, 1, 133-181 (1922) · <a href="/48.0201.01" class="nowrap">JFM 48.0201.01</a></td> </tr><tr> <td>[4]</td> <td class="space">Banjac, G.; Goulart, PJ, Tight global linear convergence rate bounds for operator splitting methods, IEEE Trans. Autom. Control, 63, 12, 4126-4139 (2018) · <a href="/1423.90175" class="nowrap">Zbl 1423.90175</a></td> </tr><tr> <td>[5]</td> <td class="space">Bauschke, HH; Combettes, PL, Convex Analysis and Monotone Operator Theory in Hilbert Spaces (2017), Berlin: Springer, Berlin · <a href="/1359.26003" class="nowrap">Zbl 1359.26003</a></td> </tr><tr> <td>[6]</td> <td class="space">Bauschke, HH; Noll, D.; Phan, HM, Linear and strong convergence of algorithms involving averaged nonexpansive operators, J. Math. Anal. Appl., 421, 1, 1-20 (2015) · <a href="/1297.65060" class="nowrap">Zbl 1297.65060</a></td> </tr><tr> <td>[7]</td> <td class="space">Bauschke, H.H., Wang, X.: Firmly nonexpansive and Kirszbraun-Valentine extensions: a constructive approach via monotone operator theory. In: Nonlinear Analysis and Optimization I: Nonlinear Analysis, pp. 55-64. American Mathematics Society (2010) · <a href="/1247.47028" class="nowrap">Zbl 1247.47028</a></td> </tr><tr> <td>[8]</td> <td class="space">Bauschke, H.H., Wang, X., Yao, L.: General resolvents for monotone operators: characterization and extension. In: Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning, and Inverse Problems, pp. 57-74. Medical Physics Publishing (2010)</td> </tr><tr> <td>[9]</td> <td class="space">Beck, A., First-Order Methods in Optimization (2017), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · <a href="/1384.65033" class="nowrap">Zbl 1384.65033</a></td> </tr><tr> <td>[10]</td> <td class="space">Bellman, R., On the theory of dynamic programming, Proc. Natl. Acad. Sci., 38, 8, 716-719 (1952) · <a href="/0047.13802" class="nowrap">Zbl 0047.13802</a></td> </tr><tr> <td>[11]</td> <td class="space">Bolte, J.; Nguyen, TP; Peypouquet, J.; Suter, BW, From error bounds to the complexity of first-order descent methods for convex functions, Math. 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