{"corpus_id":118220971,"paper_sha":"48f72a9b8748ac5e4f9b9751902ca23e658f6914","doi":null,"arxiv_id":"1207.5295","pmid":null,"pmcid":null,"mag_id":1553832572,"dblp_id":null,"acl_id":null,"title":"Directional derivatives and subdifferentials of set-valued convex functions","year":2012,"publication_date":"2012-07-23","venue":"","journal":{"name":"arXiv: Optimization and Control","pages":null,"volume":""},"journal_issn":null,"journal_title":null,"publication_types":[],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics"],"reference_count":29,"citation_count":14,"influential_citation_count":0,"is_open_access":false,"arxiv_categories":["math.OC"],"arxiv_license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":null,"s2_open_access_landing_url":null,"s2_open_access_license":null,"s2_open_access_status":null,"pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"A new directional derivative and a new subdifferential for set-valued convex functions are constructed, and a set-valued version of the so-called 'max-formula' is proven. The new concepts are used to characterize solutions of convex optimization problems with a set-valued objective. As a major tool, a residuation operation is used which acts in a space of closed convex, but not necessarily bounded subsets of a topological linear space. The residuation serves as a substitute for the inverse addition and is intimately related to the Minkowski or geometric difference of convex sets. 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