Document Zbl 1338.47076

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Algorithms</a> <a href="/?q=in%3A353825" title="Articles in this Issue">71, No. 3, 519-540 (2016)</a>. </div> <div class="abstract">Summary: We introduce and investigate the convergence properties of an inertial forward-backward-forward splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitzian operator. By making use of the product space approach, we expand it to the solving of inclusion problems involving mixtures of linearly composed and parallel-sum type monotone operators. We obtain in this way an inertial forward-backward-forward primal-dual splitting algorithm having as main characteristic the fact that in the iterative scheme all operators are accessed separately either via forward or via backward evaluations. We present also the variational case when one is interested in the solving of a primal-dual pair of convex optimization problems with complexly structured objectives, which we also illustrate by numerical experiments in image processing.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6560376">Cited in <strong>97</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47J22" title="MSC2020">47J22</a> </td> <td class="space"> Variational and other types of inclusions </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%3A65K05" title="MSC2020">65K05</a> </td> <td class="space"> Numerical mathematical programming methods </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C25" title="MSC2020">90C25</a> </td> <td class="space"> Convex programming </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amaximally+monotone+operator">maximally monotone operator</a>; <a href="/?q=ut%3Aresolvent">resolvent</a>; <a href="/?q=ut%3Asubdifferential">subdifferential</a>; <a href="/?q=ut%3Aconvex+optimization">convex optimization</a>; <a href="/?q=ut%3Ainertial+splitting+algorithm">inertial splitting algorithm</a>; <a href="/?q=ut%3Aprimal-dual+algorithm">primal-dual algorithm</a></div> <div class="software"> <h3>Software:</h3><a href="/software/9623">iPiano</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 1338.47076" data-ciurl="/ci/06560376" data-biburl="/bibtex/06560376.bib" data-amsurl="/amsrefs/06560376.bib" data-xmlurl="/xml/06560376.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06560376.pdf" title="Zbl 1338.47076 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/s11075-015-0007-5" aria-label="DOI for “An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems”" title="10.1007/s11075-015-0007-5">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1402.5291"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">Alvarez, F.: On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control. Optim. 38(4), 1102-1119 (2000) · <a href="/0954.34053" class="nowrap">Zbl 0954.34053</a> · <a href="https://doi.org/10.1137/S0363012998335802" class="nowrap">doi:10.1137/S0363012998335802</a></td> </tr><tr> <td>[2]</td> <td class="space">Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14(3), 773-782 (2004) · <a href="/1079.90096" class="nowrap">Zbl 1079.90096</a> · <a href="https://doi.org/10.1137/S1052623403427859" class="nowrap">doi:10.1137/S1052623403427859</a></td> </tr><tr> <td>[3]</td> <td class="space">Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3-11 (2001) · <a href="/0991.65056" class="nowrap">Zbl 0991.65056</a> · <a href="https://doi.org/10.1023/A:1011253113155" class="nowrap">doi:10.1023/A:1011253113155</a></td> </tr><tr> <td>[4]</td> <td class="space">Attouch, H., Briceño-Arias, L.M., Combettes, P.L.: A parallel splitting method for coupled monotone inclusions. SIAM J. Control. Optim. 48(5), 3246-3270 (2010) · <a href="/1218.47089" class="nowrap">Zbl 1218.47089</a> · <a href="https://doi.org/10.1137/090754297" class="nowrap">doi:10.1137/090754297</a></td> </tr><tr> <td>[5]</td> <td class="space">Attouch, H., Théra, M.: A general duality principle for the sum of two operators. J. Convex Anal. 3, 1-24 (1996) · <a href="/0861.47028" class="nowrap">Zbl 0861.47028</a></td> </tr><tr> <td>[6]</td> <td class="space">Bauschke, H.H., Combettes, P.L.: Convex Analysis Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011) · <a href="/1218.47001" class="nowrap">Zbl 1218.47001</a> · <a href="https://doi.org/10.1007/978-1-4419-9467-7" class="nowrap">doi:10.1007/978-1-4419-9467-7</a></td> </tr><tr> <td>[7]</td> <td class="space">Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples. Cambridge University Press, Cambridge (2010) · <a href="/1191.26001" class="nowrap">Zbl 1191.26001</a> · <a href="https://doi.org/10.1017/CBO9781139087322" class="nowrap">doi:10.1017/CBO9781139087322</a></td> </tr><tr> <td>[8]</td> <td class="space">Bot, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin Heidelberg (2010) · <a href="/1190.90002" class="nowrap">Zbl 1190.90002</a> · <a href="https://doi.org/10.1007/978-3-642-04900-2" class="nowrap">doi:10.1007/978-3-642-04900-2</a></td> </tr><tr> <td>[9]</td> <td class="space">Bot, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61(1), 35-65 (2012) · <a href="/1246.46062" class="nowrap">Zbl 1246.46062</a> · <a href="https://doi.org/10.1080/02331934.2010.505649" class="nowrap">doi:10.1080/02331934.2010.505649</a></td> </tr><tr> <td>[10]</td> <td class="space">Bot, R.I., Csetnek, E.R., Heinrich, A.: A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators. SIAM J. Optim. 23(4), 2011-2036 (2013) · <a href="/1314.47102" class="nowrap">Zbl 1314.47102</a> · <a href="https://doi.org/10.1137/12088255X" class="nowrap">doi:10.1137/12088255X</a></td> </tr><tr> <td>[11]</td> <td class="space">Bot, R.I., Csetnek, E.R., Heinrich, A., Hendrich, C.: On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems. Math. Program. 150(2), 251-279 (2015) · <a href="/1312.47081" class="nowrap">Zbl 1312.47081</a> · <a href="https://doi.org/10.1007/s10107-014-0766-0" class="nowrap">doi:10.1007/s10107-014-0766-0</a></td> </tr><tr> <td>[12]</td> <td class="space">Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472-487 (2015) · <a href="/1338.65145" class="nowrap">Zbl 1338.65145</a> · <a href="https://doi.org/10.1016/j.amc.2015.01.017" class="nowrap">doi:10.1016/j.amc.2015.01.017</a></td> </tr><tr> <td>[13]</td> <td class="space">Bot, R.I., Csetnek, E.R., László, S.: An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. arXiv:1410.0641 · <a href="/1338.90311" class="nowrap">Zbl 1338.90311</a></td> </tr><tr> <td>[14]</td> <td class="space">Bot, R.I., Hendrich, C.: Convergence analysis for a primal-dual monotone + skew splitting algorithm with applications to total variation minimization. J. Math. Imaging Vision 49(3), 551-568 (2014) · <a href="/1302.65142" class="nowrap">Zbl 1302.65142</a> · <a href="https://doi.org/10.1007/s10851-013-0486-8" class="nowrap">doi:10.1007/s10851-013-0486-8</a></td> </tr><tr> <td>[15]</td> <td class="space">Bot, R.I., Hendrich, C.: A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators. SIAM J. Optim. 23(4), 2541-2565 (2013) · <a href="/1295.47066" class="nowrap">Zbl 1295.47066</a> · <a href="https://doi.org/10.1137/120901106" class="nowrap">doi:10.1137/120901106</a></td> </tr><tr> <td>[16]</td> <td class="space">Briceño-Arias, L.M., Combettes, P.L.: A monotone + skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 21(4), 1230-1250 (2011) · <a href="/1239.47053" class="nowrap">Zbl 1239.47053</a> · <a href="https://doi.org/10.1137/10081602X" class="nowrap">doi:10.1137/10081602X</a></td> </tr><tr> <td>[17]</td> <td class="space">Cabot, A., Frankel, P.: Asymptotics for some proximal-like method involving inertia and memory aspects. Set-Valued Var. Anal. 19, 59-74 (2011) · <a href="/1214.65037" class="nowrap">Zbl 1214.65037</a> · <a href="https://doi.org/10.1007/s11228-010-0140-1" class="nowrap">doi:10.1007/s11228-010-0140-1</a></td> </tr><tr> <td>[18]</td> <td class="space">Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20(1-2), 89-97 (2004) · <a href="/1366.94048" class="nowrap">Zbl 1366.94048</a></td> </tr><tr> <td>[19]</td> <td class="space">Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision 40(1), 120-145 (2011) · <a href="/1255.68217" class="nowrap">Zbl 1255.68217</a> · <a href="https://doi.org/10.1007/s10851-010-0251-1" class="nowrap">doi:10.1007/s10851-010-0251-1</a></td> </tr><tr> <td>[20]</td> <td class="space">Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5-6), 475-504 (2004) · <a href="/1153.47305" class="nowrap">Zbl 1153.47305</a> · <a href="https://doi.org/10.1080/02331930412331327157" class="nowrap">doi:10.1080/02331930412331327157</a></td> </tr><tr> <td>[21]</td> <td class="space">Combettes, P.L., Pesquet, J.-C.: Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal. 20(2), 307-330 (2012) · <a href="/1284.47043" class="nowrap">Zbl 1284.47043</a> · <a href="https://doi.org/10.1007/s11228-011-0191-y" class="nowrap">doi:10.1007/s11228-011-0191-y</a></td> </tr><tr> <td>[22]</td> <td class="space">Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158 (2), 460-479 (2013) · <a href="/1272.90110" class="nowrap">Zbl 1272.90110</a> · <a href="https://doi.org/10.1007/s10957-012-0245-9" class="nowrap">doi:10.1007/s10957-012-0245-9</a></td> </tr><tr> <td>[23]</td> <td class="space">Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Company, Amsterdam (1976) · <a href="/0322.90046" class="nowrap">Zbl 0322.90046</a></td> </tr><tr> <td>[24]</td> <td class="space">Maingé, P.-E.: Convergence theorems for inertial KM-type algorithms. J. Comput. Appl. Math. 219, 223-236 (2008) · <a href="/1156.65054" class="nowrap">Zbl 1156.65054</a> · <a href="https://doi.org/10.1016/j.cam.2007.07.021" class="nowrap">doi:10.1016/j.cam.2007.07.021</a></td> </tr><tr> <td>[25]</td> <td class="space">Maingé, P.-E., Moudafi, A.: Convergence of new inertial proximal methods for dc programming. SIAM J. Optim. 19(1), 397-413 (2008) · <a href="/1158.49034" class="nowrap">Zbl 1158.49034</a> · <a href="https://doi.org/10.1137/060655183" class="nowrap">doi:10.1137/060655183</a></td> </tr><tr> <td>[26]</td> <td class="space">Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155, 447-454 (2003) · <a href="/1027.65077" class="nowrap">Zbl 1027.65077</a> · <a href="https://doi.org/10.1016/S0377-0427(02)00906-8" class="nowrap">doi:10.1016/S0377-0427(02)00906-8</a></td> </tr><tr> <td>[27]</td> <td class="space">Ochs, P., Chen, Y., Brox, T., Pock, T.: iPiano: Inertial proximal algorithm for non-convex optimization. SIAM J. Imag. Sci. 7(2), 1388-1419 (2014) · <a href="/1296.90094" class="nowrap">Zbl 1296.90094</a> · <a href="https://doi.org/10.1137/130942954" class="nowrap">doi:10.1137/130942954</a></td> </tr><tr> <td>[28]</td> <td class="space">Pesquet, J.-C., Pustelnik, N.: A parallel inertial proximal optimization method. Pac. J. Optim. 8(2), 273-305 (2012) · <a href="/1259.47080" class="nowrap">Zbl 1259.47080</a></td> </tr><tr> <td>[29]</td> <td class="space">Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33(1), 209-216 (1970) · <a href="/0199.47101" class="nowrap">Zbl 0199.47101</a> · <a href="https://doi.org/10.2140/pjm.1970.33.209" class="nowrap">doi:10.2140/pjm.1970.33.209</a></td> </tr><tr> <td>[30]</td> <td class="space">Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14(5), 877-898 (1976) · <a href="/0358.90053" class="nowrap">Zbl 0358.90053</a> · <a href="https://doi.org/10.1137/0314056" class="nowrap">doi:10.1137/0314056</a></td> </tr><tr> <td>[31]</td> <td class="space">Simons, S.: From Hahn-Banach to Monotonicity. Springer, Berlin (2008) · <a href="/1131.47050" class="nowrap">Zbl 1131.47050</a></td> </tr><tr> <td>[32]</td> <td class="space">Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control. Optim. 29(1), 119-138 (1991) · <a href="/0737.90048" class="nowrap">Zbl 0737.90048</a> · <a href="https://doi.org/10.1137/0329006" class="nowrap">doi:10.1137/0329006</a></td> </tr><tr> <td>[33]</td> <td class="space">Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control. Optim. 38(2), 431-446 (2000) · <a href="/0997.90062" class="nowrap">Zbl 0997.90062</a> · <a href="https://doi.org/10.1137/S0363012998338806" class="nowrap">doi:10.1137/S0363012998338806</a></td> </tr><tr> <td>[34]</td> <td class="space">Vu, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38(3), 667-681 (2013) · <a href="/1284.47045" class="nowrap">Zbl 1284.47045</a> · <a href="https://doi.org/10.1007/s10444-011-9254-8" class="nowrap">doi:10.1007/s10444-011-9254-8</a></td> </tr><tr> <td>[35]</td> <td class="space">Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002) · <a href="/1023.46003" class="nowrap">Zbl 1023.46003</a> · <a href="https://doi.org/10.1142/9789812777096" class="nowrap">doi:10.1142/9789812777096</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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