{"id":1103958,"date":"2024-11-14T10:22:10","date_gmt":"2024-11-14T18:22:10","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=1103958"},"modified":"2024-11-15T06:34:43","modified_gmt":"2024-11-15T14:34:43","slug":"bounding-the-gap-between-the-mccormick-relaxation-and-the-convex-hull-for-bilinear-functions","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/bounding-the-gap-between-the-mccormick-relaxation-and-the-convex-hull-for-bilinear-functions\/","title":{"rendered":"Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions"},"content":{"rendered":"<p>We investigate how well the graph of a bilinear function \\(b:[0,1]^n\u2192\\mathbb{R}\\) can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most c times the difference between the concave and convex envelopes. Answering a question of Luedtke, Namazifar and Linderoth, we show that this factor c cannot be bounded by a constant independent of n. More precisely, we show that for a random bilinear function b we have asymptotically almost surely \\(c\u2a7e\\sqrt{n}\/4\\). On the other hand, we prove that \\(c\u2a7d600\\sqrt{n}\\), which improves the linear upper bound proved by Luedtke, Namazifar and Linderoth. In addition, we present an alternative proof for a result of Misener, Smadbeck and Floudas characterizing functions <em>b<\/em> for which the McCormick relaxation is equal to the convex hull.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We investigate how well the graph of a bilinear function can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most c times the difference between [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"523","msr_page_range_end":"535","msr_series":"","msr_volume":"162","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":null,"msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2017-5-29","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":false,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":null,"footnotes":""},"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[246691,246907],"msr-conference":[],"msr-journal":[268542],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-1103958","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us","msr-field-of-study-computer-science","msr-field-of-study-mathematics"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2017-5-29","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"162","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":0,"msr_main_download":"","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"doi","viewUrl":"false","id":"false","title":"https:\/\/doi.org\/10.1007\/s10107-016-1031-5","label_id":"243106","label":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[],"msr-author-ordering":[{"type":"text","value":"N. Boland","user_id":0,"rest_url":false},{"type":"text","value":"Santanu S. Dey","user_id":0,"rest_url":false},{"type":"text","value":"T. Kalinowski","user_id":0,"rest_url":false},{"type":"user_nicename","value":"Marco Molinaro","user_id":42204,"rest_url":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=Marco Molinaro"},{"type":"text","value":"F. 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