{"id":1383,"date":"2023-06-01T08:34:05","date_gmt":"2023-06-01T08:34:05","guid":{"rendered":"https:\/\/nag.com\/?page_id=1383"},"modified":"2023-07-14T11:22:27","modified_gmt":"2023-07-14T11:22:27","slug":"bound-optimization-by-quadratic-approximation","status":"publish","type":"page","link":"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/","title":{"rendered":"Bound Optimization BY Quadratic Approximation"},"content":{"rendered":"\n
\n
\n
\n
\n
\n
\n \n

Bound Optimization BY Quadratic Approximation<\/h1>\n \n

n<\/i><\/span>AG Library mini-article<\/p>\n<\/div>\n\n \n <\/div>\n <\/div>\n <\/div>\n <\/div>\n <\/div>\n<\/div>\n\n\n\n

<\/div>\n\n
\n
\n
\n

Mark 23 of the n<\/i><\/span>AG Library and the n<\/i><\/span>AG Toolbox\u00a0for MATLAB\u00ae\u00a0expands the functionality provided by the Chapter for Minimizing or Maximizing a Function (E04). The BOBYQA (Bound Optimization BY Quadratic Approximation) algorithm of Prof. Mike Powell, University of Cambridge, is now available in the Library. This robust method is an easy-to-use algorithm that employs quadratic approximation and trust regions to minimize an objective subject to bound constraints. No derivatives of the objective function are required, and the solver’s efficiency is preserved for large problem sizes.<\/p>\n

As a simple example, the problem of distributing 50 points on a sphere to have maximal pairwise separation, starting from equispaced points on the equator (see also Powell (2009)), is solved using 4633 function evaluations. This compares with 16757 taken by the n<\/i><\/span>AG Nelder\u2013Mead simplex solver on the same problem (run on a machine using GCC 4.5.2, Fedora 10, four 2.00GHz dual-core Intel\u00ae Xeon\u00ae E5405 processors, 8Gb RAM).<\/p>\n

Reference: Powell M.J.D. (2009), “The BOBYQA algorithm for bound constrained optimization without derivatives”, Report DAMTP 2009\/NA06, University of Cambridge.<\/p>\n

Related information<\/h3>\n

The Price of Derivatives – Derivative-free Optimization \u00b7\u00a0read more<\/a><\/p>\n

<\/p>\n

n<\/i><\/span>AG Library Function Document<\/a><\/span>\u00a0for (e04jcc) – Bound constrained minimum, model-based algorithm, using function values only<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n

<\/div>","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"content-type":"","footnotes":""},"class_list":["post-1383","page","type-page","status-publish","hentry"],"yoast_head":"\nBound Optimization BY Quadratic Approximation - nAG<\/title>\n<meta name=\"description\" content=\"Solving the problem of distributing 50 points on a sphere to have maximal pairwise separation, starting from equispaced points on the equator.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bound Optimization BY Quadratic Approximation - nAG\" \/>\n<meta property=\"og:description\" content=\"Solving the problem of distributing 50 points on a sphere to have maximal pairwise separation, starting from equispaced points on the equator.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/\" \/>\n<meta property=\"og:site_name\" content=\"nAG\" \/>\n<meta property=\"article:modified_time\" content=\"2023-07-14T11:22:27+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/nag.com\/wp-content\/uploads\/2024\/02\/NAG-Logo-White-On-Blue.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"12770\" \/>\n\t<meta property=\"og:image:height\" content=\"4353\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:site\" content=\"@NAGTalk\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/\",\"url\":\"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/\",\"name\":\"Bound Optimization BY Quadratic Approximation - nAG\",\"isPartOf\":{\"@id\":\"https:\/\/nag.com\/#website\"},\"datePublished\":\"2023-06-01T08:34:05+00:00\",\"dateModified\":\"2023-07-14T11:22:27+00:00\",\"description\":\"Solving the problem of distributing 50 points on a sphere to have maximal pairwise separation, starting from equispaced points on the equator.\",\"breadcrumb\":{\"@id\":\"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/nag.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Bound Optimization BY Quadratic Approximation\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/nag.com\/#website\",\"url\":\"https:\/\/nag.com\/\",\"name\":\"NAG\",\"description\":\"Robust, trusted numerical software and computational expertise.\",\"publisher\":{\"@id\":\"https:\/\/nag.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/nag.com\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/nag.com\/#organization\",\"name\":\"Numerical Algorithms Group\",\"alternateName\":\"NAG\",\"url\":\"https:\/\/nag.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/nag.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/nag.com\/wp-content\/uploads\/2023\/11\/NAG-Logo.png\",\"contentUrl\":\"https:\/\/nag.com\/wp-content\/uploads\/2023\/11\/NAG-Logo.png\",\"width\":1244,\"height\":397,\"caption\":\"Numerical Algorithms Group\"},\"image\":{\"@id\":\"https:\/\/nag.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/x.com\/NAGTalk\",\"https:\/\/www.linkedin.com\/company\/nag\/\",\"https:\/\/www.youtube.com\/user\/NumericalAlgorithms\",\"https:\/\/github.com\/numericalalgorithmsgroup\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Bound Optimization BY Quadratic Approximation - nAG","description":"Solving the problem of distributing 50 points on a sphere to have maximal pairwise separation, starting from equispaced points on the equator.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/","og_locale":"en_US","og_type":"article","og_title":"Bound Optimization BY Quadratic Approximation - nAG","og_description":"Solving the problem of distributing 50 points on a sphere to have maximal pairwise separation, starting from equispaced points on the equator.","og_url":"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/","og_site_name":"nAG","article_modified_time":"2023-07-14T11:22:27+00:00","og_image":[{"width":12770,"height":4353,"url":"https:\/\/nag.com\/wp-content\/uploads\/2024\/02\/NAG-Logo-White-On-Blue.jpg","type":"image\/jpeg"}],"twitter_card":"summary_large_image","twitter_site":"@NAGTalk","schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/","url":"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/","name":"Bound Optimization BY Quadratic Approximation - nAG","isPartOf":{"@id":"https:\/\/nag.com\/#website"},"datePublished":"2023-06-01T08:34:05+00:00","dateModified":"2023-07-14T11:22:27+00:00","description":"Solving the problem of distributing 50 points on a sphere to have maximal pairwise separation, starting from equispaced points on the equator.","breadcrumb":{"@id":"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/nag.com\/bound-optimization-by-quadratic-approximation\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/nag.com\/"},{"@type":"ListItem","position":2,"name":"Bound Optimization BY Quadratic Approximation"}]},{"@type":"WebSite","@id":"https:\/\/nag.com\/#website","url":"https:\/\/nag.com\/","name":"NAG","description":"Robust, trusted numerical software and computational expertise.","publisher":{"@id":"https:\/\/nag.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/nag.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/nag.com\/#organization","name":"Numerical Algorithms Group","alternateName":"NAG","url":"https:\/\/nag.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/nag.com\/#\/schema\/logo\/image\/","url":"https:\/\/nag.com\/wp-content\/uploads\/2023\/11\/NAG-Logo.png","contentUrl":"https:\/\/nag.com\/wp-content\/uploads\/2023\/11\/NAG-Logo.png","width":1244,"height":397,"caption":"Numerical Algorithms Group"},"image":{"@id":"https:\/\/nag.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/x.com\/NAGTalk","https:\/\/www.linkedin.com\/company\/nag\/","https:\/\/www.youtube.com\/user\/NumericalAlgorithms","https:\/\/github.com\/numericalalgorithmsgroup"]}]}},"_links":{"self":[{"href":"https:\/\/nag.com\/wp-json\/wp\/v2\/pages\/1383","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/nag.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/nag.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/nag.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/nag.com\/wp-json\/wp\/v2\/comments?post=1383"}],"version-history":[{"count":4,"href":"https:\/\/nag.com\/wp-json\/wp\/v2\/pages\/1383\/revisions"}],"predecessor-version":[{"id":65614,"href":"https:\/\/nag.com\/wp-json\/wp\/v2\/pages\/1383\/revisions\/65614"}],"wp:attachment":[{"href":"https:\/\/nag.com\/wp-json\/wp\/v2\/media?parent=1383"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}