{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T21:14:28Z","timestamp":1767215668537,"version":"3.48.0"},"reference-count":12,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Elastic-net regularization, as a variational method, demonstrates enhanced stability compared to classical\n                    <jats:inline-formula id=\"j_cmam-2024-0176_ineq_9999\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msub>\n                            <m:mi mathvariant=\"normal\">\u2113<\/m:mi>\n                            <m:mn>1<\/m:mn>\n                          <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2024-0176_eq_0317.png\"\/>\n                        <jats:tex-math>{\\ell_{1}}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    sparsity regularization, making it suitable for addressing ill-conditioned problems. However, conventional elastic-net regularization is typically limited to linear equations. In this paper, we extend the elastic-net regularization method to nonlinear problems. We investigate the well-posedness of this regularization and demonstrate that it serves as a sparsity regularization approach. The iterative soft thresholding algorithm, commonly used for classical\n                    <jats:inline-formula id=\"j_cmam-2024-0176_ineq_9998\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msub>\n                            <m:mi mathvariant=\"normal\">\u2113<\/m:mi>\n                            <m:mn>1<\/m:mn>\n                          <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2024-0176_eq_0317.png\"\/>\n                        <jats:tex-math>{\\ell_{1}}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    sparsity regularization, features a straightforward structure and is easy to implement. We show that, under widely accepted conditions regarding the nonlinearity of the function\n                    <jats:italic>F<\/jats:italic>\n                    , this algorithm is effective in solving the elastic-net regularization for nonlinear ill-conditioned equations. Our numerical results highlight the efficiency of the proposed method.\n                  <\/jats:p>","DOI":"10.1515\/cmam-2024-0176","type":"journal-article","created":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T00:01:19Z","timestamp":1762819279000},"page":"127-146","source":"Crossref","is-referenced-by-count":0,"title":["Nonlinear Elastic-Net Regularization and Its Iterative Soft Thresholding Algorithm"],"prefix":"10.1515","volume":"26","author":[{"given":"Yu","family":"Tian","sequence":"first","affiliation":[{"name":"Department of Mathematics , 47820 Northeast Forestry University , Harbin 150040 , P. R. China"}]},{"given":"Liang","family":"Ding","sequence":"additional","affiliation":[{"name":"Department of Mathematics , 47820 Northeast Forestry University , Harbin 150040 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2025,11,10]]},"reference":[{"key":"2025123121115018027_j_cmam-2024-0176_ref_001","doi-asserted-by":"crossref","unstructured":"A.  Beck and Y. C.  Eldar,\nSparsity constrained nonlinear optimization: Optimality conditions and algorithms,\nSIAM J. Optim. 23 (2013), no. 3, 1480\u20131509.","DOI":"10.1137\/120869778"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_002","doi-asserted-by":"crossref","unstructured":"T.  Bonesky, K.  Bredies, D. A.  Lorenz and P.  Maass,\nA generalized conditional gradient method for nonlinear operator equations with sparsity constraints,\nInverse Problems 23 (2007), no. 5, 2041\u20132058.","DOI":"10.1088\/0266-5611\/23\/5\/014"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_003","doi-asserted-by":"crossref","unstructured":"K.  Bredies, D. A.  Lorenz and P.  Maass,\nA generalized conditional gradient method and its connection to an iterative shrinkage method,\nComput. Optim. Appl. 42 (2009), no. 2, 173\u2013193.","DOI":"10.1007\/s10589-007-9083-3"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_004","doi-asserted-by":"crossref","unstructured":"L.  Ding and W.  Han,\n\n                  \n                     \n                        \n                           \u03b1\n                           \u2062\n                           \n                              \u2113\n                              1\n                           \n                        \n                     \n                     \n                     \\alpha\\ell_{1}\n                  \n               -\n                  \n                     \n                        \n                           \u03b2\n                           \u2062\n                           \n                              \u2113\n                              2\n                           \n                        \n                     \n                     \n                     \\beta\\ell_{2}\n                  \n                regularization for sparse recovery,\nInverse Problems 35 (2019), no. 12, Article ID 125009.","DOI":"10.1088\/1361-6420\/ab34b5"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_005","doi-asserted-by":"crossref","unstructured":"L.  Ding and W.  Han,\n\n                  \n                     \n                        \n                           \n                              \u03b1\n                              \u2062\n                              \n                                 \ud835\udc9e\n                                 1\n                              \n                           \n                           -\n                           \n                              \u03b2\n                              \u2062\n                              \n                                 \ud835\udc9e\n                                 2\n                              \n                           \n                        \n                     \n                     \n                     \\alpha\\mathcal{C}_{1}-\\beta\\mathcal{C}_{2}\n                  \n                sparsity regularization for nonlinear ill-posed problems,\nJ. Comput. Appl. Math. 450 (2024), Article ID 115987.","DOI":"10.1016\/j.cam.2024.115987"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_006","doi-asserted-by":"crossref","unstructured":"M. A. T.  Figueiredo, R. D.  Nowak and S. J.  Wright,\nGradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,\nIEEE J. Selected Topics Signal Process. 1 (2009), 586\u2013597.","DOI":"10.1109\/JSTSP.2007.910281"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_007","doi-asserted-by":"crossref","unstructured":"P. C.  Hansen,\nRegularization tools version 3.0 for Matlab 5.2,\nNumer. Algorithms 20 (1999), 195\u2013196.","DOI":"10.1023\/A:1019160018978"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_008","doi-asserted-by":"crossref","unstructured":"B.  Jin and P.  Maass,\nSparsity regularization for parameter identification problems,\nInverse Problems 28 (2012), no. 12, Article ID 123001.","DOI":"10.1088\/0266-5611\/28\/12\/123001"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_009","unstructured":"O.  Scherzer, M.  Grasmair, H.  Grossauer, M.  Haltmeier and F.  Lenzen,\nVariational Methods in Imaging,\nAppl. Math. Sci. 167,\nSpringer, New York, 2009."},{"key":"2025123121115018027_j_cmam-2024-0176_ref_010","doi-asserted-by":"crossref","unstructured":"Y.  Wang, L.  Ren, Z.  Zhang, G.  Lin and C.  Xu,\nSparsity-promoting elastic net method with rotations for high-dimensional nonlinear inverse problem,\nComput. Methods Appl. Mech. Engrg. 345 (2019), 263\u2013282.","DOI":"10.1016\/j.cma.2018.10.040"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_011","doi-asserted-by":"crossref","unstructured":"R.  Wilson, A. D.  Calway and E. R. S.  Pearson,\nA generalized wavelet transform for Fourier analysis: The multiresolution Fourier transform and its application to image and audio signal analysis,\nIEEE Trans. Inform. Theory 38 (1992), no. 2, 674\u2013690.","DOI":"10.1109\/18.119730"},{"key":"2025123121115018027_j_cmam-2024-0176_ref_012","doi-asserted-by":"crossref","unstructured":"H.  Zou and T.  Hastie,\nRegularization and variable selection via the elastic net,\nJ. R. Stat. Soc. Ser. B Stat. Methodol. 67 (2005), no. 2, 301\u2013320.","DOI":"10.1111\/j.1467-9868.2005.00503.x"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0176\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0176\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T21:12:22Z","timestamp":1767215542000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0176\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,10]]},"references-count":12,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,1]]},"published-print":{"date-parts":[[2026,1,1]]}},"alternative-id":["10.1515\/cmam-2024-0176"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2024-0176","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2025,11,10]]}}}