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By integrating variable smoothing techniques with first-order methods, we propose a variable smoothing alternating proximal gradient algorithm that features flexible parameter choices for step sizes and smoothing levels. Under mild assumptions, we establish that the iteration complexity to reach an\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varepsilon $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03b5<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -approximate stationary point is\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {O}(\\varepsilon ^{-3})$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\u03b5<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mo>-<\/mml:mo>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . The proposed algorithm is evaluated on sparse signal recovery and image denoising problems. Numerical experiments demonstrate its effectiveness and superiority over existing algorithms.\n                  <\/jats:p>","DOI":"10.1007\/s10957-026-02966-8","type":"journal-article","created":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T04:07:12Z","timestamp":1774498032000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Variable Smoothing Alternating Proximal Gradient Algorithm for Coupled Composite Optimization"],"prefix":"10.1007","volume":"209","author":[{"given":"Xian-Jun","family":"Long","sequence":"first","affiliation":[]},{"given":"Kang","family":"Zeng","sequence":"additional","affiliation":[]},{"given":"Gao-Xi","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8074-6675","authenticated-orcid":false,"given":"Minh N.","family":"Dao","sequence":"additional","affiliation":[]},{"given":"Zai-Yun","family":"Peng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,26]]},"reference":[{"issue":"2","key":"2966_CR1","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1287\/moor.1100.0449","volume":"35","author":"H Attouch","year":"2010","unstructured":"Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-\u0141ojasiewicz inequality. 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