Document Zbl 07971683

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Comput. Math.</a> <a href="/?q=in%3A522178" title="Articles in this Issue">51, No. 1, Paper No. 4, 29 p. (2025)</a>. </div> <div class="abstract">Summary: The generalized Lanczos trust-region (GLTR) method is one of the most popular approaches for solving large-scale trust-region subproblem (TRS). In Jia and Wang, <span class="zbmathjax-textit">SIAM J. Optim., 31, 887–914</span> 2021. Z. Jia et al. considered the convergence of this method and established some <span class="zbmathjax-textit">a priori</span> error bounds on the residual and the Lagrange multiplier. In this paper, we revisit the convergence of the GLTR method and try to improve these bounds. First, we establish a sharper upper bound on the residual. Second, we present a <span class="zbmathjax-textit">non-asymptotic</span> bound for the convergence of the Lagrange multiplier and define a factor that plays an important role in the convergence of the Lagrange multiplier. Third, we revisit the convergence of the Krylov subspace method for the cubic regularization variant of the trust-region subproblem and substantially improve the convergence result established in Jia et al., <span class="zbmathjax-textit">SIAM J. Matrix Anal. Appl. 43 (2022), pp. 812–839</span> 2022 on the multiplier. Numerical experiments demonstrate the effectiveness of our theoretical results.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65F15" title="MSC2020">65F15</a> </td> <td class="space"> Numerical computation of eigenvalues and eigenvectors of matrices </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C26" title="MSC2020">90C26</a> </td> <td class="space"> Nonconvex programming, global optimization </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Atrust-region+subproblem">trust-region subproblem</a>; <a href="/?q=ut%3Ageneralized+Lanczos+trust-region+%28GLTR%29+method">generalized Lanczos trust-region (GLTR) method</a>; <a href="/?q=ut%3AKrylov+subspace">Krylov subspace</a>; <a href="/?q=ut%3Acubic+regularization">cubic regularization</a>; <a href="/?q=ut%3Aeasy+case">easy case</a></div> <div class="software"> <h3>Software:</h3><a href="/software/36311">HSL-VF05</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 07971683" data-ciurl="/ci/07971683" data-biburl="/bibtex/07971683.bib" data-amsurl="/amsrefs/07971683.bib" data-xmlurl="/xml/07971683.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07971683.pdf" title="Zbl 07971683 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/s10444-024-10217-5" aria-label="DOI for “On convergence of the generalized Lanczos trust-region method for trust-region subproblems”" title="10.1007/s10444-024-10217-5">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2207.12674"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">Adachi, S.; Iwata, S.; Nakatsukasa, Y.; Takeda, A., Solving the trust-region subproblem by a generalized eigenvalue problem, SIAM J. Optim., 27, 269-291, 2017; Adachi, S.; Iwata, S.; Nakatsukasa, Y.; Takeda, A., Solving the trust-region subproblem by a generalized eigenvalue problem, SIAM J. Optim., 27, 269-291, 2017 · <a href="/1359.49009" class="nowrap">Zbl 1359.49009</a> · <a href="https://doi.org/10.1137/16M1058200" class="nowrap">doi:10.1137/16M1058200</a></td> </tr><tr> <td>[2]</td> <td class="space">Bianconcini, T.; Liuzzi, G.; Morini, B.; Sciandrone, M., On the use of iterative methods in cubic regularization for unconstrained optimization, Comput. Optim. Appl., 60, 35-57, 2015; Bianconcini, T.; Liuzzi, G.; Morini, B.; Sciandrone, M., On the use of iterative methods in cubic regularization for unconstrained optimization, Comput. Optim. Appl., 60, 35-57, 2015 · <a href="/1308.90166" class="nowrap">Zbl 1308.90166</a> · <a href="https://doi.org/10.1007/s10589-014-9672-x" class="nowrap">doi:10.1007/s10589-014-9672-x</a></td> </tr><tr> <td>[3]</td> <td class="space">Carmon, Y.; Duchi, J., First-order method for nonconvex quadratic minimization, SIAM Rev., 62, 395-436, 2020; Carmon, Y.; Duchi, J., First-order method for nonconvex quadratic minimization, SIAM Rev., 62, 395-436, 2020 · <a href="/1459.65082" class="nowrap">Zbl 1459.65082</a> · <a href="https://doi.org/10.1137/20M1321759" class="nowrap">doi:10.1137/20M1321759</a></td> </tr><tr> <td>[4]</td> <td class="space">Carmon, Y., Duchi, J.: Analysis of Krylov subspace solutions of regularized nonconvex quadratic problems. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems, pp. 10728-10738 (2018)</td> </tr><tr> <td>[5]</td> <td class="space">Cartis, C., Gould, N., Toint, P.: Adaptive cubic regularisation methods for unconstrained optimization. Part I: Motivation, convergence and numerical results, Math. Program. 127, 245-295 (2011) · <a href="/1229.90192" class="nowrap">Zbl 1229.90192</a></td> </tr><tr> <td>[6]</td> <td class="space">Cartis, C.; Gould, N.; Lange, M., On monotonic estimates of the norm of the minimizers of regularized quadratic functions in Krylov spaces, BIT Numer. Math., 60, 583-589, 2020; Cartis, C.; Gould, N.; Lange, M., On monotonic estimates of the norm of the minimizers of regularized quadratic functions in Krylov spaces, BIT Numer. Math., 60, 583-589, 2020 · <a href="/1448.90069" class="nowrap">Zbl 1448.90069</a> · <a href="https://doi.org/10.1007/s10543-019-00791-2" class="nowrap">doi:10.1007/s10543-019-00791-2</a></td> </tr><tr> <td>[7]</td> <td class="space">Conn, A.; Gould, N.; Toint, P., Trust-region methods, 2000, Philadelphia, PA: SIAM, Philadelphia, PA; Conn, A.; Gould, N.; Toint, P., Trust-region methods, 2000, Philadelphia, PA: SIAM, Philadelphia, PA · <a href="/0958.65071" class="nowrap">Zbl 0958.65071</a> · <a href="https://doi.org/10.1137/1.9780898719857" class="nowrap">doi:10.1137/1.9780898719857</a></td> </tr><tr> <td>[8]</td> <td class="space">Erway, J.; Gill, P.; Griffin, J., Iterative methods for finding a trust-region step, SIAM J. Optim., 20, 1110-1131, 2009; Erway, J.; Gill, P.; Griffin, J., Iterative methods for finding a trust-region step, SIAM J. Optim., 20, 1110-1131, 2009 · <a href="/1189.49049" class="nowrap">Zbl 1189.49049</a> · <a href="https://doi.org/10.1137/070708494" class="nowrap">doi:10.1137/070708494</a></td> </tr><tr> <td>[9]</td> <td class="space">Gander, W.; Golub, GH; von Matt, U., A constrained eigenvalue problem, Linear Algebra Appl., 114, 815-839, 1989; Gander, W.; Golub, GH; von Matt, U., A constrained eigenvalue problem, Linear Algebra Appl., 114, 815-839, 1989 · <a href="/0666.15006" class="nowrap">Zbl 0666.15006</a> · <a href="https://doi.org/10.1016/0024-3795(89)90494-1" class="nowrap">doi:10.1016/0024-3795(89)90494-1</a></td> </tr><tr> <td>[10]</td> <td class="space">Golub, GH; von Matt, U., Quadratically constrained least squares and quadratic problems, Numer. Math., 59, 561-580, 1991; Golub, GH; von Matt, U., Quadratically constrained least squares and quadratic problems, Numer. Math., 59, 561-580, 1991 · <a href="/0745.65029" class="nowrap">Zbl 0745.65029</a> · <a href="https://doi.org/10.1007/BF01385796" class="nowrap">doi:10.1007/BF01385796</a></td> </tr><tr> <td>[11]</td> <td class="space">Gould, N.; Lucidi, S.; Roma, M.; Toint, P., Solving the trust-region subproblem using the Lanczos method, SIAM J. Optim., 9, 504-525, 1999; Gould, N.; Lucidi, S.; Roma, M.; Toint, P., Solving the trust-region subproblem using the Lanczos method, SIAM J. Optim., 9, 504-525, 1999 · <a href="/1047.90510" class="nowrap">Zbl 1047.90510</a> · <a href="https://doi.org/10.1137/S1052623497322735" class="nowrap">doi:10.1137/S1052623497322735</a></td> </tr><tr> <td>[12]</td> <td class="space">Gould, N.; Robinson, D.; Thorne, H., On solving trust-region and other regularised subproblems in optimization, Math. Program. Comput., 2, 21-57, 2010; Gould, N.; Robinson, D.; Thorne, H., On solving trust-region and other regularised subproblems in optimization, Math. Program. Comput., 2, 21-57, 2010 · <a href="/1193.65098" class="nowrap">Zbl 1193.65098</a> · <a href="https://doi.org/10.1007/s12532-010-0011-7" class="nowrap">doi:10.1007/s12532-010-0011-7</a></td> </tr><tr> <td>[13]</td> <td class="space">Gould, N.; Simoncini, V., Error estimates for iterative algorithms for minimizing regularized quadratic subproblems, Optim. Methods Softw., 35, 304-328, 2020; Gould, N.; Simoncini, V., Error estimates for iterative algorithms for minimizing regularized quadratic subproblems, Optim. Methods Softw., 35, 304-328, 2020 · <a href="/1428.90160" class="nowrap">Zbl 1428.90160</a> · <a href="https://doi.org/10.1080/10556788.2019.1670177" class="nowrap">doi:10.1080/10556788.2019.1670177</a></td> </tr><tr> <td>[14]</td> <td class="space">Hager, W., Minimizing a quadratic over a sphere, SIAM J. Optim., 12, 188-208, 2001; Hager, W., Minimizing a quadratic over a sphere, SIAM J. Optim., 12, 188-208, 2001 · <a href="/1058.90045" class="nowrap">Zbl 1058.90045</a> · <a href="https://doi.org/10.1137/S1052623499356071" class="nowrap">doi:10.1137/S1052623499356071</a></td> </tr><tr> <td>[15]</td> <td class="space">Hager, W.; Krylyuk, Y., Graph partitioning and continuous quadratic programming, SIAM J. Discrete Math., 12, 500-523, 1999; Hager, W.; Krylyuk, Y., Graph partitioning and continuous quadratic programming, SIAM J. Discrete Math., 12, 500-523, 1999 · <a href="/0972.90086" class="nowrap">Zbl 0972.90086</a> · <a href="https://doi.org/10.1137/S0895480199335829" class="nowrap">doi:10.1137/S0895480199335829</a></td> </tr><tr> <td>[16]</td> <td class="space">Horn, R.; Johnson, C., Matrix analysis, 2013, Cambridge, UK: Cambridge University Press, Cambridge, UK; Horn, R.; Johnson, C., Matrix analysis, 2013, Cambridge, UK: Cambridge University Press, Cambridge, UK · <a href="/1267.15001" class="nowrap">Zbl 1267.15001</a></td> </tr><tr> <td>[17]</td> <td class="space">Jia, X.; Liang, X.; Shen, C.; Zhang, L., Solving the cubic regularization model by a nested restarting Lanczos method, SIAM J. Matrix Anal. Appl., 43, 812-839, 2022; Jia, X.; Liang, X.; Shen, C.; Zhang, L., Solving the cubic regularization model by a nested restarting Lanczos method, SIAM J. Matrix Anal. Appl., 43, 812-839, 2022 · <a href="/1489.90183" class="nowrap">Zbl 1489.90183</a> · <a href="https://doi.org/10.1137/21M1436324" class="nowrap">doi:10.1137/21M1436324</a></td> </tr><tr> <td>[18]</td> <td class="space">Jia, Z., The convergence of generalized Lanczos methods for large unsymmetric eigenproblems, SIAM J. Matrix Anal. Appl., 16, 843-862, 1995; Jia, Z., The convergence of generalized Lanczos methods for large unsymmetric eigenproblems, SIAM J. Matrix Anal. Appl., 16, 843-862, 1995 · <a href="/0834.65021" class="nowrap">Zbl 0834.65021</a> · <a href="https://doi.org/10.1137/S0895479893246753" class="nowrap">doi:10.1137/S0895479893246753</a></td> </tr><tr> <td>[19]</td> <td class="space">Jia, Z.; Wang, F., The convergence of the generalized Lanczos trust-region method for the trust-region subproblem, SIAM J. Optim., 31, 887-914, 2021; Jia, Z.; Wang, F., The convergence of the generalized Lanczos trust-region method for the trust-region subproblem, SIAM J. Optim., 31, 887-914, 2021 · <a href="/1462.90083" class="nowrap">Zbl 1462.90083</a> · <a href="https://doi.org/10.1137/19M1279691" class="nowrap">doi:10.1137/19M1279691</a></td> </tr><tr> <td>[20]</td> <td class="space">Lieder, F., Solving large-scale cubic regularization by a generalized eigenvalue problem, SIAM J. Optim., 30, 3345-3358, 2020; Lieder, F., Solving large-scale cubic regularization by a generalized eigenvalue problem, SIAM J. Optim., 30, 3345-3358, 2020 · <a href="/1458.90589" class="nowrap">Zbl 1458.90589</a> · <a href="https://doi.org/10.1137/19M1291388" class="nowrap">doi:10.1137/19M1291388</a></td> </tr><tr> <td>[21]</td> <td class="space">Lukšan, L.; Matonoha, C.; Vlček, J., On Lagrange multipliers of trust-region subproblems, BIT, 48, 763-768, 2008; Lukšan, L.; Matonoha, C.; Vlček, J., On Lagrange multipliers of trust-region subproblems, BIT, 48, 763-768, 2008 · <a href="/1163.90027" class="nowrap">Zbl 1163.90027</a> · <a href="https://doi.org/10.1007/s10543-008-0197-5" class="nowrap">doi:10.1007/s10543-008-0197-5</a></td> </tr><tr> <td>[22]</td> <td class="space">Moré, J.; Sorensen, D., Computing a trust region step, SIAM J. Sci. Statist. Comput., 4, 553-572, 1983; Moré, J.; Sorensen, D., Computing a trust region step, SIAM J. Sci. Statist. Comput., 4, 553-572, 1983 · <a href="/0551.65042" class="nowrap">Zbl 0551.65042</a> · <a href="https://doi.org/10.1137/0904038" class="nowrap">doi:10.1137/0904038</a></td> </tr><tr> <td>[23]</td> <td class="space">Nesterov, Y.; Polyak, B., Cubic regularization of Newton method and its global performance, Math. Program., 108, 177-205, 2006; Nesterov, Y.; Polyak, B., Cubic regularization of Newton method and its global performance, Math. Program., 108, 177-205, 2006 · <a href="/1142.90500" class="nowrap">Zbl 1142.90500</a> · <a href="https://doi.org/10.1007/s10107-006-0706-8" class="nowrap">doi:10.1007/s10107-006-0706-8</a></td> </tr><tr> <td>[24]</td> <td class="space">Nocedal, J.; Writht, S., Numerical optimization, 2006, New York: Springer, New York; Nocedal, J.; Writht, S., Numerical optimization, 2006, New York: Springer, New York · <a href="/1104.65059" class="nowrap">Zbl 1104.65059</a></td> </tr><tr> <td>[25]</td> <td class="space">Rendl, F.; Wolkowicz, H., A semidefinite framework for trust region subproblems with applications to large scale minimization, Math. Program., 77, 273-299, 1997; Rendl, F.; Wolkowicz, H., A semidefinite framework for trust region subproblems with applications to large scale minimization, Math. Program., 77, 273-299, 1997 · <a href="/0888.90137" class="nowrap">Zbl 0888.90137</a> · <a href="https://doi.org/10.1007/BF02614438" class="nowrap">doi:10.1007/BF02614438</a></td> </tr><tr> <td>[26]</td> <td class="space">Rojas, M.; Santos, S.; Sorensen, D., A new matrix-free algorithm for the large-scale trust-region subproblem, SIAM J. Optim., 11, 611-646, 2000; Rojas, M.; Santos, S.; Sorensen, D., A new matrix-free algorithm for the large-scale trust-region subproblem, SIAM J. Optim., 11, 611-646, 2000 · <a href="/0994.65067" class="nowrap">Zbl 0994.65067</a> · <a href="https://doi.org/10.1137/S105262349928887X" class="nowrap">doi:10.1137/S105262349928887X</a></td> </tr><tr> <td>[27]</td> <td class="space">Rojas, M.; Santos, S.; Sorensen, D., Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularization, ACM Trans. Math. Software, 34, 1-28, 2008; Rojas, M.; Santos, S.; Sorensen, D., Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularization, ACM Trans. Math. Software, 34, 1-28, 2008 · <a href="/1291.65177" class="nowrap">Zbl 1291.65177</a> · <a href="https://doi.org/10.1145/1326548.1326553" class="nowrap">doi:10.1145/1326548.1326553</a></td> </tr><tr> <td>[28]</td> <td class="space">Saad, Y., Iterative methods for sparse linear systems, 2003, Philadelphia, PA: SIAM, Philadelphia, PA; Saad, Y., Iterative methods for sparse linear systems, 2003, Philadelphia, PA: SIAM, Philadelphia, PA · <a href="/1031.65046" class="nowrap">Zbl 1031.65046</a> · <a href="https://doi.org/10.1137/1.9780898718003" class="nowrap">doi:10.1137/1.9780898718003</a></td> </tr><tr> <td>[29]</td> <td class="space">Sorensen, D., Newton’s method with a model trust region modification, SIAM J. Numer. Anal., 19, 409-426, 1982; Sorensen, D., Newton’s method with a model trust region modification, SIAM J. Numer. Anal., 19, 409-426, 1982 · <a href="/0483.65039" class="nowrap">Zbl 0483.65039</a> · <a href="https://doi.org/10.1137/0719026" class="nowrap">doi:10.1137/0719026</a></td> </tr><tr> <td>[30]</td> <td class="space">Sorensen, D., Minimization of a large-scale quadratic function subject to a spherical constraint, SIAM J. Optim., 7, 141-161, 1997; Sorensen, D., Minimization of a large-scale quadratic function subject to a spherical constraint, SIAM J. Optim., 7, 141-161, 1997 · <a href="/0878.65044" class="nowrap">Zbl 0878.65044</a> · <a href="https://doi.org/10.1137/S1052623494274374" class="nowrap">doi:10.1137/S1052623494274374</a></td> </tr><tr> <td>[31]</td> <td class="space">Steihaug, T., The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20, 626-637, 1983; Steihaug, T., The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20, 626-637, 1983 · <a href="/0518.65042" class="nowrap">Zbl 0518.65042</a> · <a href="https://doi.org/10.1137/0720042" class="nowrap">doi:10.1137/0720042</a></td> </tr><tr> <td>[32]</td> <td class="space">Tao, P.; An, L., A DC optimization algorithm for solving the trust-region subproblem, SIAM J. Optim., 8, 476-505, 1998; Tao, P.; An, L., A DC optimization algorithm for solving the trust-region subproblem, SIAM J. Optim., 8, 476-505, 1998 · <a href="/0913.65054" class="nowrap">Zbl 0913.65054</a> · <a href="https://doi.org/10.1137/S1052623494274313" class="nowrap">doi:10.1137/S1052623494274313</a></td> </tr><tr> <td>[33]</td> <td class="space">Toint, P.; Duff, I., Towards an efficient sparsity exploiting Newton method for minimization, Sparse Matrices and Their Uses, 57-88, 1981, London: Academic Press, London; Toint, P.; Duff, I., Towards an efficient sparsity exploiting Newton method for minimization, Sparse Matrices and Their Uses, 57-88, 1981, London: Academic Press, London · <a href="/0463.65045" class="nowrap">Zbl 0463.65045</a></td> </tr><tr> <td>[34]</td> <td class="space">Zhang, L.; Shen, C.; Li, R., On the generalized Lanczos trust-region method, SIAM J. Optim., 27, 2110-2142, 2017; Zhang, L.; Shen, C.; Li, R., On the generalized Lanczos trust-region method, SIAM J. Optim., 27, 2110-2142, 2017 · <a href="/1380.90210" class="nowrap">Zbl 1380.90210</a> · <a href="https://doi.org/10.1137/16M1095056" class="nowrap">doi:10.1137/16M1095056</a></td> </tr><tr> <td>[35]</td> <td class="space">Zhang, L., Shen, C.: A nested Lanczos method for the trust-region subproblem, SIAM J. Sci. Comput. 40, A2005-A2032 (2018) · <a href="/1402.90113" class="nowrap">Zbl 1402.90113</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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