{"id":"https://openalex.org/W7140552112","doi":"https://doi.org/10.1007/s10957-026-02966-8","title":"Variable Smoothing Alternating Proximal Gradient Algorithm for Coupled Composite Optimization","display_name":"Variable Smoothing Alternating Proximal Gradient Algorithm for Coupled Composite Optimization","publication_year":2026,"publication_date":"2026-03-26","ids":{"openalex":"https://openalex.org/W7140552112","doi":"https://doi.org/10.1007/s10957-026-02966-8"},"language":"en","primary_location":{"id":"doi:10.1007/s10957-026-02966-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-026-02966-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-026-02966-8.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10957-026-02966-8.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5130712942","display_name":"Xian-Jun Long","orcid":null},"institutions":[{"id":"https://openalex.org/I145581781","display_name":"Chongqing Technology and Business University","ror":"https://ror.org/05hqf1284","country_code":"CN","type":"education","lineage":["https://openalex.org/I145581781"]},{"id":"https://openalex.org/I50632499","display_name":"Chongqing University of Technology","ror":"https://ror.org/04vgbd477","country_code":"CN","type":"education","lineage":["https://openalex.org/I50632499"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xian-Jun Long","raw_affiliation_strings":["Chongqing Key Laboratory of Statistical Intelligent Computing and Monitoring, Chongqing Technology and Business University, Chongqing, 400067, People\u2019s Republic of China","School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People\u2019s Republic of China"],"affiliations":[{"raw_affiliation_string":"Chongqing Key Laboratory of Statistical Intelligent Computing and Monitoring, Chongqing Technology and Business University, Chongqing, 400067, People\u2019s Republic of China","institution_ids":["https://openalex.org/I145581781","https://openalex.org/I50632499"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People\u2019s Republic of China","institution_ids":["https://openalex.org/I145581781"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5130640546","display_name":"Kang Zeng","orcid":null},"institutions":[{"id":"https://openalex.org/I145581781","display_name":"Chongqing Technology and Business University","ror":"https://ror.org/05hqf1284","country_code":"CN","type":"education","lineage":["https://openalex.org/I145581781"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Kang Zeng","raw_affiliation_strings":["School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People\u2019s Republic of China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People\u2019s Republic of China","institution_ids":["https://openalex.org/I145581781"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5114023920","display_name":"Gao-Xi Li","orcid":null},"institutions":[{"id":"https://openalex.org/I145581781","display_name":"Chongqing Technology and Business University","ror":"https://ror.org/05hqf1284","country_code":"CN","type":"education","lineage":["https://openalex.org/I145581781"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Gao-Xi Li","raw_affiliation_strings":["School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People\u2019s Republic of China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People\u2019s Republic of China","institution_ids":["https://openalex.org/I145581781"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5121817406","display_name":"Minh N. Dao","orcid":null},"institutions":[{"id":"https://openalex.org/I82951845","display_name":"RMIT University","ror":"https://ror.org/04ttjf776","country_code":"AU","type":"education","lineage":["https://openalex.org/I82951845"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Minh N. Dao","raw_affiliation_strings":["School of Science, RMIT University, Melbourne, VIC, 3000, Australia"],"affiliations":[{"raw_affiliation_string":"School of Science, RMIT University, Melbourne, VIC, 3000, Australia","institution_ids":["https://openalex.org/I82951845"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5130637295","display_name":"Zai-Yun Peng","orcid":null},"institutions":[{"id":"https://openalex.org/I120825670","display_name":"Yunnan Normal University","ror":"https://ror.org/00sc9n023","country_code":"CN","type":"education","lineage":["https://openalex.org/I120825670"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zai-Yun Peng","raw_affiliation_strings":["School of Mathematics, Yunnan Normal University, Kunming, Yunnan, 650092, People\u2019s Republic of China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, Yunnan Normal University, Kunming, Yunnan, 650092, People\u2019s Republic of China","institution_ids":["https://openalex.org/I120825670"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5130712942"],"corresponding_institution_ids":["https://openalex.org/I145581781","https://openalex.org/I50632499"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.71578639,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"209","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.8255000114440918,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.8255000114440918,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.0835999995470047,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.06520000100135803,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/smoothing","display_name":"Smoothing","score":0.9046000242233276},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.5911999940872192},{"id":"https://openalex.org/keywords/proximal-gradient-methods","display_name":"Proximal Gradient Methods","score":0.5745000243186951},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.534600019454956},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.47530001401901245},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.47130000591278076},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.44909998774528503},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.44699999690055847},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4383000135421753}],"concepts":[{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.9046000242233276},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7472000122070312},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6111000180244446},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.5911999940872192},{"id":"https://openalex.org/C10494615","wikidata":"https://www.wikidata.org/wiki/Q17086765","display_name":"Proximal Gradient Methods","level":4,"score":0.5745000243186951},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.534600019454956},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5012000203132629},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47530001401901245},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.47130000591278076},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.44909998774528503},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.44699999690055847},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4383000135421753},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.4226999878883362},{"id":"https://openalex.org/C163294075","wikidata":"https://www.wikidata.org/wiki/Q581861","display_name":"Noise reduction","level":2,"score":0.4194999933242798},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.36480000615119934},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.3580000102519989},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.3492000102996826},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3407000005245209},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.32690000534057617},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.31929999589920044},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.3111000061035156},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.28600001335144043},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.27630001306533813},{"id":"https://openalex.org/C2987595161","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Optimization algorithm","level":2,"score":0.25290000438690186},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.25279998779296875}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10957-026-02966-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-026-02966-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-026-02966-8.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Optimization Theory and Applications","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10957-026-02966-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-026-02966-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-026-02966-8.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G391238517","display_name":null,"funder_award_id":", and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4020255992","display_name":null,"funder_award_id":"Project","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4292976619","display_name":null,"funder_award_id":"12271067","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5939423041","display_name":null,"funder_award_id":"Technology","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7173093016","display_name":"MultiTree: Completing the Library of Language Relationships","funder_award_id":"1227106","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7865710446","display_name":null,"funder_award_id":"DP230101749","funder_id":"https://openalex.org/F4320334704","funder_display_name":"Australian Research Council"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8524746985","display_name":null,"funder_award_id":"NSCQ-","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322685","display_name":"Chongqing Technology and Business University","ror":"https://ror.org/05hqf1284"},{"id":"https://openalex.org/F4320334704","display_name":"Australian Research Council","ror":"https://ror.org/05mmh0f86"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7140552112.pdf","grobid_xml":"https://content.openalex.org/works/W7140552112.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W205960364","https://openalex.org/W1820680895","https://openalex.org/W1965125844","https://openalex.org/W1967138577","https://openalex.org/W1984481261","https://openalex.org/W2000462146","https://openalex.org/W2027982384","https://openalex.org/W2029463628","https://openalex.org/W2086336156","https://openalex.org/W2092663520","https://openalex.org/W2129732816","https://openalex.org/W2140514146","https://openalex.org/W2142506503","https://openalex.org/W2241435520","https://openalex.org/W2265744594","https://openalex.org/W2549127690","https://openalex.org/W2743758565","https://openalex.org/W2791154158","https://openalex.org/W2803106003","https://openalex.org/W2972561294","https://openalex.org/W3093944120","https://openalex.org/W3129118870","https://openalex.org/W4250955649","https://openalex.org/W4281387205","https://openalex.org/W4362636876","https://openalex.org/W4386749862","https://openalex.org/W4388293171","https://openalex.org/W4399612589","https://openalex.org/W4406141814","https://openalex.org/W4412865127","https://openalex.org/W4415420918","https://openalex.org/W4415931467"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2],"paper,":[3],"we":[4,38,61],"consider":[5],"a":[6,20,27,40],"broad":[7],"class":[8],"of":[9],"nonconvex":[10],"and":[11,55,108,117],"nonsmooth":[12,21],"optimization":[13],"problems,":[14],"where":[15],"one":[16],"objective":[17],"component":[18],"is":[19,79,102],"weakly":[22],"convex":[23],"function":[24],"composed":[25],"with":[26,35],"linear":[28],"operator.":[29],"By":[30],"integrating":[31],"variable":[32,41],"smoothing":[33,42,56],"techniques":[34],"first-order":[36],"methods,":[37],"propose":[39],"alternating":[43],"proximal":[44],"gradient":[45],"algorithm":[46,101],"that":[47,63],"features":[48],"flexible":[49],"parameter":[50],"choices":[51],"for":[52],"step":[53],"sizes":[54],"levels.":[57],"Under":[58],"mild":[59],"assumptions,":[60],"establish":[62],"the":[64],"iteration":[65],"complexity":[66],"to":[67],"reach":[68],"an":[69],"$$\\varepsilon":[70],"$$":[71],"<mml:math":[72,83],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[73,84],"<mml:mi>\u03b5</mml:mi>":[74,89],"</mml:math>":[75,97],"-approximate":[76],"stationary":[77],"point":[78],"$$\\mathcal":[80],"{O}(\\varepsilon":[81],"^{-3})$$":[82],"<mml:mrow>":[85,90],"<mml:mi>O</mml:mi>":[86],"<mml:mo>(</mml:mo>":[87],"<mml:msup>":[88],"<mml:mo>-</mml:mo>":[91],"<mml:mn>3</mml:mn>":[92],"</mml:mrow>":[93,96],"</mml:msup>":[94],"<mml:mo>)</mml:mo>":[95],".":[98],"The":[99],"proposed":[100],"evaluated":[103],"on":[104],"sparse":[105],"signal":[106],"recovery":[107],"image":[109],"denoising":[110],"problems.":[111],"Numerical":[112],"experiments":[113],"demonstrate":[114],"its":[115],"effectiveness":[116],"superiority":[118],"over":[119],"existing":[120],"algorithms.":[121]},"counts_by_year":[],"updated_date":"2026-04-18T07:56:08.524223","created_date":"2026-03-27T00:00:00"}
