{"id":4834,"date":"2022-12-19T19:35:45","date_gmt":"2022-12-19T11:35:45","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=4834"},"modified":"2022-12-19T21:53:50","modified_gmt":"2022-12-19T13:53:50","slug":"duality","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=4834","title":{"rendered":"Duality"},"content":{"rendered":"\n<p>$$min\\ f_0(x), x \\in \\mathbb{R}^n$$<\/p>\n\n\n\n<p>$$s.t. f_i(x)\\leq 0, \\text{for i from 1 to m}$$<\/p>\n\n\n\n<p>$$ \\quad h_i(x)=0, \\text{for i from 1 to q}$$<\/p>\n\n\n\n<ul><li>That is, in Lagrangian form,<\/li><\/ul>\n\n\n\n<p>$$L(x,\\lambda,\\gamma)=f_0((x)+\\sum \\lambda_i f_i (x) +\\sum \\gamma_i h_i(x)$$<\/p>\n\n\n\n<p>$$ \\min_{x} \\max_{\\lambda,\\gamma} L(x,\\lambda, \\gamma) $$<\/p>\n\n\n\n<p>$$s.t. \\lambda \\geq 0$$<\/p>\n\n\n\n<ul><li>The Duality Problem is,<\/li><\/ul>\n\n\n\n<p>$$g(\\lambda, \\gamma) = \\min_{x} L(x,\\lambda, \\gamma)$$<\/p>\n\n\n\n<p>$$\\max_{\\lambda, \\gamma} g(\\lambda, \\gamma) $$<\/p>\n\n\n\n<p>$$s.t. \\nabla_x L(x,\\lambda, \\gamma)=0$$<\/p>\n\n\n\n<p>$$\\quad \\lambda \\geq 0$$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Why Duality?<\/h4>\n\n\n\n<p>We change the original problem in to the <strong>duality<\/strong>, which becomes a convexity optimisation problem. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Convexity Optimisation<\/h4>\n\n\n\n<ul><li>The object function is convex. (or the negative of a concavity)<\/li><li>The feasible set is a convex set.<\/li><\/ul>\n\n\n\n<p><strong>See further study.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>$$min\\ f_0(x), x \\in \\mathbb{R}^n$$ $$s.t. f_i(x)\\leq 0, \\text{for i from 1 to m}$$ $$ \\quad h_i(x)=0, \\text{for i from 1 to q}$$ That is, in Lagrangian form, $$L(x,\\lambda,\\gamma)=f_0((x)+\\sum \\lambda_i f_i (x) +\\sum \\gamma_i h_i(x)$$ $$ \\min_{x} \\max_{\\lambda,\\gamma} L(x,\\lambda, \\gamma) $$ $$s.t. \\lambda \\geq 0$$ The Duality Problem is, $$g(\\lambda, \\gamma) = \\min_{x} L(x,\\lambda, \\gamma)$$ &hellip; <a href=\"https:\/\/fanyuzhao.com\/?p=4834\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Duality<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,8,18],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4834"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4834"}],"version-history":[{"count":13,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4834\/revisions"}],"predecessor-version":[{"id":4865,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4834\/revisions\/4865"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4834"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4834"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4834"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}