Document Zbl 1229.90123

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Anal.</a> <a href="/?q=in%3A271152" title="Articles in this Issue">18, No. 3-4, 373-404 (2010)</a>. </div> <div class="abstract">Summary: In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenable to solution by current methods but they feature Fenchel-Moreau-Rockafellar dual problems that can be solved by forward-backward splitting. The proposed algorithm produces simultaneously a sequence converging weakly to a dual solution, and a sequence converging strongly to the primal solution. Our framework is shown to capture and extend several existing duality-based signal recovery methods and to be applicable to a variety of new problems beyond their scope.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5837852">Cited in <strong>41</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A49N15" title="MSC2020">49N15</a> </td> <td class="space"> Duality theory (optimization) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A12" title="MSC2020">94A12</a> </td> <td class="space"> Signal theory (characterization, reconstruction, filtering, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A08" title="MSC2020">94A08</a> </td> <td class="space"> Image processing (compression, reconstruction, etc.) in information and communication theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aconvex+optimization">convex optimization</a>; <a href="/?q=ut%3Adenoising">denoising</a>; <a href="/?q=ut%3Adictionary">dictionary</a>; <a href="/?q=ut%3ADykstra-like+algorithm">Dykstra-like algorithm</a>; <a href="/?q=ut%3Aduality">duality</a>; <a href="/?q=ut%3Aforward-backward+splitting">forward-backward splitting</a>; <a href="/?q=ut%3Aimage+reconstruction">image reconstruction</a>; <a href="/?q=ut%3Aimage+restoration">image restoration</a>; <a href="/?q=ut%3Ainverse+problem">inverse problem</a>; <a href="/?q=ut%3Asignal+recovery">signal recovery</a>; <a href="/?q=ut%3Aprimal-dual+algorithm">primal-dual algorithm</a>; <a href="/?q=ut%3Aproximity+operator">proximity operator</a>; <a href="/?q=ut%3Atotal+variation">total variation</a></div> <div class="software"> <h3>Software:</h3><a href="/software/8365">SPGL1</a>; <a href="/software/20466">TwIST</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 1229.90123" data-ciurl="/ci/05837852" data-biburl="/bibtex/05837852.bib" data-amsurl="/amsrefs/05837852.bib" data-xmlurl="/xml/05837852.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05837852.pdf" title="Zbl 1229.90123 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/s11228-010-0147-7" aria-label="DOI for “Dualization of signal recovery problems”" title="10.1007/s11228-010-0147-7">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/0907.0436"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">Amar, M., Bellettini, G.: A notion of total variation depending on a metric with discontinuous coefficients. 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