Document Zbl 1257.90042

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Program. Comput.</a> <a href="/?q=in%3A286219" title="Articles in this Issue">3, No. 3, 165-218 (2011)</a>. </div> <div class="abstract">Summary: This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, $\parallel Wx \parallel_1$ where $W$ is arbitrary, or a combination thereof. In addition, the paper introduces a number of technical contributions such as a novel continuation scheme and a novel approach for controlling the step size, and applies results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6110466">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A6110466">Cited in <strong>135</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C05" title="MSC2020">90C05</a> </td> <td class="space"> Linear programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C06" title="MSC2020">90C06</a> </td> <td class="space"> Large-scale problems in mathematical programming </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%3A62J07" title="MSC2020">62J07</a> </td> <td class="space"> Ridge regression; shrinkage estimators (Lasso) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aoptimal+first-order+methods">optimal first-order methods</a>; <a href="/?q=ut%3ANesterov%27s+accelerated+descent+algorithms">Nesterov’s accelerated descent algorithms</a>; <a href="/?q=ut%3Aproximal+algorithms">proximal algorithms</a>; <a href="/?q=ut%3Aconic+duality">conic duality</a>; <a href="/?q=ut%3Asmoothing+by+conjugation">smoothing by conjugation</a>; <a href="/?q=ut%3ADantzig+selector">Dantzig selector</a>; <a href="/?q=ut%3ALASSO">LASSO</a>; <a href="/?q=ut%3Anuclear-norm+minimization">nuclear-norm minimization</a></div> <div class="software"> <h3>Software:</h3><a href="/software/4019">PROPACK</a>; <a href="/software/8366">ParNes</a>; <a href="/software/5148">PDCO</a>; <a href="/software/12430">L1-MAGIC</a>; <a href="/software/6576">NESTA</a>; <a href="/software/4594">CVX</a>; <a href="/software/8159">glmnet</a>; <a href="/software/4618">Mosek</a>; <a href="/software/20470">L1TestPack</a>; <a href="/software/12839">TFOCS</a>; <a href="/software/8365">SPGL1</a>; <a href="/software/12218">FPC_AS</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 1257.90042" data-ciurl="/ci/06110466" data-biburl="/bibtex/06110466.bib" data-amsurl="/amsrefs/06110466.bib" data-xmlurl="/xml/06110466.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06110466.pdf" title="Zbl 1257.90042 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/s12532-011-0029-5" aria-label="DOI for “Templates for convex cone problems with applications to sparse signal recovery”" title="10.1007/s12532-011-0029-5">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1009.2065"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">Afonso, M.V., Bioucas-Dias, J.M., Figueiredo, M.A.T.: An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems. 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