{"id":2351,"date":"2016-11-20T16:22:07","date_gmt":"2016-11-20T08:22:07","guid":{"rendered":"https:\/\/boweihe.me\/?p=2351"},"modified":"2016-11-20T16:22:07","modified_gmt":"2016-11-20T08:22:07","slug":"%e7%bf%bb%e8%af%91%ef%bc%9a%e6%ac%a1%e6%a2%af%e5%ba%a6%e4%bb%a5%e5%8f%8a%e4%b8%80%e9%98%b6%e6%9c%80%e4%bc%98%e6%80%a7%e6%9d%a1%e4%bb%b6subgradient-and-first-order-optimality-condition","status":"publish","type":"post","link":"https:\/\/dayandcarrot.space\/?p=2351","title":{"rendered":"\u7ffb\u8bd1\uff1a\u6b21\u68af\u5ea6\u4ee5\u53ca\u4e00\u9636\u6700\u4f18\u6027\u6761\u4ef6(Subgradient and First-order Optimality Condition)"},"content":{"rendered":"<p>\u672c\u6587\u662f\u5bf9\u6587\u7ae0<a href=\"http:\/\/www.stats.ox.ac.uk\/~lienart\/blog_opti_basics.html\" target=\"_blank\" rel=\"noopener noreferrer\">Basics of Convex Analysis<\/a>\u7684\u90e8\u5206\u7ffb\u8bd1\uff0c\u82e5\u672c\u6587\u5bf9\u60a8\u7684\u4efb\u4f55\u6743\u76ca\u9020\u6210\u4fb5\u72af\uff0c\u8bf7\u8054\u7cfb\u6211\u3002<br \/>\nThis article is a partial translation of\u00a0<a href=\"http:\/\/www.stats.ox.ac.uk\/~lienart\/blog_opti_basics.html\" target=\"_blank\" rel=\"noopener noreferrer\"><em>Basics of Convex Analysis<\/em><\/a>, if this infringes any of your rights, please contact me.<\/p>\n<hr \/>\n<h3>\u6b21\u68af\u5ea6\u4ee5\u53ca\u4e00\u9636\u6700\u4f18\u6027\u6761\u4ef6<\/h3>\n<p><span style=\"color: #3366ff;\">\u82e5\u4e0b\u8ff0\u4e0d\u7b49\u5f0f\u6210\u7acb\uff1a<\/span><br \/>\n<span style=\"color: #3366ff;\">[latex]f(z) \\geqslant{f(x)} + &lt;z-x, y&gt;, \\forall z \\in X.[\/latex]<\/span><br \/>\n<span style=\"color: #3366ff;\">\u5219\u6211\u4eec\u8bf4[latex]y\\in X[\/latex]\u662f\u51fd\u6570[latex]f \\in \\Gamma _0(X)[\/latex]\u5728[latex]x \\in X[\/latex]\u70b9\u4e0a\u7684\u4e00\u4e2a\u6b21\u68af\u5ea6\uff0c\u5e76\u4e14\u5c5e\u4e8e[latex]f[\/latex]\u5728\u8be5\u70b9(\u7528[latex]\\partial f(x)[\/latex]\u8868\u793a)\u7684\u4e00\u4e2a\u6b21\u5fae\u5206\u3002<\/span><br \/>\n\u4e0a\u8ff0\u4e0d\u7b49\u5f0f\u8868\u660e\u51fd\u6570[latex]f[\/latex]\u7684\u56fe\u50cf(graph)\u662f\u7531\u4e0d\u7b49\u5f0f\u53f3\u4fa7\u5b9a\u4e49\u7684\u8d85\u5e73\u9762\u6240(hyperplane)\u652f\u6491\u7684\u3002\u4e00\u4e2a\u6b21\u68af\u5ea6\u56e0\u6b64\u4e5f\u662f\u8fd9\u4f17\u591a\u652f\u6301\u8d85\u5e73\u9762\u5176\u4e2d\u4e00\u4e2a\u7684\u201c\u659c\u7387(slope)\u201d\u3002\u5982\u679c\u8be5\u51fd\u6570\u5728\u70b9[latex]x[\/latex]\u5904\u53ef\u5fae\uff0c\u5219\u8fd9\u6837\u7684\u6b21\u68af\u5ea6\u6709\u4e14\u4ec5\u6709\u4e00\u4e2a\uff08\u5373\u6807\u51c6\u68af\u5ea6\uff09\uff0c\u4e0e\u6b64\u5bf9\u5e94\u5730\uff0c\u4e5f\u53ea\u6709\u4e00\u4e2a\u652f\u6301\u8d85\u5e73\u9762\u3002\u76f8\u5bf9\u5730\uff0c\u5982\u679c\u4e00\u4e2a\u51fd\u6570\u5728\u70b9[latex]x[\/latex]\u5904\u4e0d\u53ef\u8c13\uff08\u6bd4\u5982\uff0c\u5728[latex]x[\/latex]\u5904\u6709\u4e2a\u626d\u66f2\uff09\u90a3\u4e48\u5c31\u80fd\u6709\u65e0\u7a77\u591a\u4e2a\u652f\u6301\u8d85\u5e73\u9762\uff0c\u5e76\u4e14\u76f8\u5bf9\u5e94\u5730\uff0c\u5728\u8be5\u70b9\u7684\u6b21\u5fae\u5206\u662f\u4e00\u4e2a\u8fde\u7eed\u7684\u6b21\u68af\u5ea6\u7684\u96c6\u5408\u3002<br \/>\n\u4e00\u4e2a\u5178\u578b\u7684\u4f8b\u5b50\u662f\u7edd\u5bf9\u503c\u51fd\u6570[latex]f(x)=|x|[\/latex]\uff0c\u5b83\u57280\u70b9\u662f\u4e0d\u53ef\u5bfc\u7684\u3002\u4f46\u5728\u8fd9\u4e2a\u70b9\u4e0a\uff0c\u5b83\u53ef\u4ee5\u7531[latex]l(x)=\\alpha x, \\alpha \\in [-1,1][\/latex]\u7ec4\u6210\u7684\u6240\u6709\u76f4\u7ebf\u652f\u6301\u3002\u8fd9\u4e2a\u96c6\u5408\u5373\u8be5\u51fd\u6570\u57280\u70b9\u7684\u6b21\u5fae\u5206\uff0c\u7528[latex]\\partial f(0)[\/latex]\u8868\u793a\u3002<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-large\" src=\"http:\/\/www.stats.ox.ac.uk\/~lienart\/_figs\/ex_subgrad_plot1_g.png\" alt=\"\" width=\"363\" height=\"244\" \/><br \/>\n\u51fd\u6570[latex]f(x)=|x|[\/latex]\u7684\u6f14\u793a\uff08\u7c97\u7ebf\u8868\u793a\uff09\u4ee5\u53ca\u5176\u4e2d\u4e24\u4e2a\u652f\u6301\u76f4\u7ebf\uff08\u865a\u7ebf\u8868\u793a\uff09\u3002\u8fd9\u4e24\u6761\u652f\u6301\u76f4\u7ebf\u90fd\u6709\u6b21\u5fae\u5206[latex]\\partial f(0)[\/latex]\u4e2d\u7684\u659c\u7387\u3002\u6ce8\u610f\uff0c\u90a3\u6761\u6c34\u5e73\u7ebf\u4e5f\u662f\u652f\u6301\u8d85\u5e73\u9762\u4e4b\u4e00\uff0c\u8868\u660e[latex]0 \\in \\partial f(0)[\/latex]\u3002\u5e76\u4e14\u56e0\u6b64\u7531\u4e00\u9636\u6761\u4ef6(\u4e0b\u6587\u5b9a\u4e49)\uff0c\u8fd9\u4e2a\u51fd\u6570\u5728\u539f\u70b9\u6709\u4e00\u4e2a\u6781\u5c0f\u503c\u3002<br \/>\n\u73b0\u5728\uff0c\u901a\u8fc7\u5b9a\u4e49\u975e\u9650\u5236\u95ee\u9898\u4e2d\u7684\u4e00\u4e2a\u6700\u4f18\u70b9[latex]x^{\\#}[\/latex]\uff0c\u5fc5\u6709[latex]f(z) \\geqslant{f(x^{\\#}) + &lt;z-x, 0&gt;}, \\forall z \\in x [\/latex]\uff0c\u5e76\u4e14\u56e0\u6b640\u5fc5\u987b\u662f\u51fd\u6570[latex]f[\/latex]\u5728[latex]x^{\\#}[\/latex]\u70b9\u5904\u7684\u4e00\u4e2a\u5b50\u68af\u5ea6\u3002<br \/>\n<span style=\"color: #3366ff;\">\u8fd9\u5c31\u662f\u4e00\u9636\u6700\u4f18\u6027\u6761\u4ef6(FOC)\uff1a<\/span><br \/>\n<span style=\"color: #3366ff;\">[latex]x^{\\#} \\in arg min_{x}{f(x)} \\Longleftrightarrow 0 \\in \\partial f(x^{\\#})[\/latex]<\/span><br \/>\n\u5982\u679c\u6211\u4eec\u5c06\u6b21\u5fae\u5206\u770b\u505a\u4e00\u4e2a\u8fd0\u7b97\u7b26\u90a3\u4e48\uff0c\u76f4\u89c2\u5730\uff0c\u5bfb\u627e\u6781\u5c0f\u53ef\u4ee5\u770b\u505a\u201c\u9006\u8f6c\u201d\u6b21\u5fae\u5206\u5e76\u8ba1\u7b97\u5b83\u5728\u70b90\u7684\u503c\u7684\u8fc7\u7a0b\uff0c\u5373[latex]x^{\\#}=(\\partial f)^{-1} (0)[\/latex]\u3002\u6211\u4eec\u7a0d\u540e\u518d\u8fdb\u4e00\u6b65\u4ecb\u7ecd\uff0c\u4f46\u8fd9\u4e2a\u9006\u8f6c\u6b21\u5fae\u5206\u8fd0\u7b97\u7b26\u7684\u601d\u8def\u662f\u975e\u5e38\u91cd\u8981\u7684\u3002<br \/>\n\u5728\u7ee7\u7eed\u4e4b\u524d\uff0c\u6709\u5fc5\u8981\u63d0\u4e00\u4e0b\uff08\u5e76\u4e14\u8fd9\u5e76\u4e0d\u96be\u7406\u89e3\uff09\u4e0b\u8ff0\u5305\u542b\u5173\u7cfb\u5bf9\u4e8e\u6b21\u5fae\u5206\u7684\u548c\u662f\u6210\u7acb\u7684\uff1a<br \/>\n[latex]\\sum_{i}{\\partial f_i} \\subseteq \\partial \\sum_{i}{f_i} [\/latex]<br \/>\n\u5bf9\u5173\u6ce8\u7684\u5927\u90e8\u5206\u95ee\u9898\u800c\u8a00\uff0c\u4e0a\u8ff0\u5173\u7cfb\u53ef\u4ee5\u5f3a\u5316\u4e3a\u76f8\u7b49\u7684\u5173\u7cfb\uff0c \u4f46\u6ce8\u610f\u5982\u679c[latex]0 \\in \\sum_{i}{\\partial f_i (x^\\dagger)}[\/latex]\uff0c\u5219\u4e0a\u8ff0\u5305\u542b\u5173\u7cfb\u610f\u5473\u7740[latex]0 \\in \\partial \\sum_{i}{f_i (x^\\dagger)}[\/latex]\uff0c\u8fd9\u5bf9\u4e8e\u8bc1\u660e[latex]x^\\dagger[\/latex]\u662f\u4e2a\u6781\u5c0f\u503c\u800c\u8a00\uff08\u8fd9\u6b63\u662f\u6211\u4eec\u6700\u611f\u5174\u8da3\u4e4b\u5904\uff09\u8db3\u77e3\u3002<\/p>\n<hr \/>\n<h2><\/h2>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u6587\u662f\u5bf9\u6587\u7ae0Basics of Convex Analysis\u7684\u90e8\u5206\u7ffb\u8bd1\uff0c\u82e5\u672c\u6587\u5bf9\u60a8\u7684\u4efb\u4f55\u6743\u76ca\u9020\u6210\u4fb5\u72af\uff0c\u8bf7\u8054\u7cfb\u6211\u3002 This article is a partial translation of\u00a0Basics of Convex Analysis, if this infringes any of your rights, please contact me. \u6b21\u68af\u5ea6\u4ee5\u53ca\u4e00\u9636\u6700\u4f18\u6027\u6761\u4ef6 \u82e5\u4e0b\u8ff0\u4e0d\u7b49\u5f0f\u6210\u7acb\uff1a [latex]f(z) \\geqslant{f(x)} + &lt;z-x, y&gt;, \\forall z \\in X.[\/latex] \u5219\u6211\u4eec\u8bf4[latex]y\\in X[\/latex]\u662f\u51fd\u6570[latex]f \\in \\Gamma _0(X)[\/latex]\u5728[latex]x \\in X[\/latex]\u70b9\u4e0a\u7684\u4e00\u4e2a\u6b21\u68af\u5ea6\uff0c\u5e76\u4e14\u5c5e\u4e8e[latex]f[\/latex]\u5728\u8be5\u70b9(\u7528[latex]\\partial f(x)[\/latex]\u8868\u793a)\u7684\u4e00\u4e2a\u6b21\u5fae\u5206\u3002 \u4e0a\u8ff0\u4e0d\u7b49\u5f0f\u8868\u660e\u51fd\u6570[latex]f[\/latex]\u7684\u56fe\u50cf(graph)\u662f\u7531\u4e0d\u7b49\u5f0f\u53f3\u4fa7\u5b9a\u4e49\u7684\u8d85\u5e73\u9762\u6240(hyperplane)\u652f\u6491\u7684\u3002\u4e00\u4e2a\u6b21\u68af\u5ea6\u56e0\u6b64\u4e5f\u662f\u8fd9\u4f17\u591a\u652f\u6301\u8d85\u5e73\u9762\u5176\u4e2d\u4e00\u4e2a\u7684\u201c\u659c\u7387(slope)\u201d\u3002\u5982\u679c\u8be5\u51fd\u6570\u5728\u70b9[latex]x[\/latex]\u5904\u53ef\u5fae\uff0c\u5219\u8fd9\u6837\u7684\u6b21\u68af\u5ea6\u6709\u4e14\u4ec5\u6709\u4e00\u4e2a\uff08\u5373\u6807\u51c6\u68af\u5ea6\uff09\uff0c\u4e0e\u6b64\u5bf9\u5e94\u5730\uff0c\u4e5f\u53ea\u6709\u4e00\u4e2a\u652f\u6301\u8d85\u5e73\u9762\u3002\u76f8\u5bf9\u5730\uff0c\u5982\u679c\u4e00\u4e2a\u51fd\u6570\u5728\u70b9[latex]x[\/latex]\u5904\u4e0d\u53ef\u8c13\uff08\u6bd4\u5982\uff0c\u5728[latex]x[\/latex]\u5904\u6709\u4e2a\u626d\u66f2\uff09\u90a3\u4e48\u5c31\u80fd\u6709\u65e0\u7a77\u591a\u4e2a\u652f\u6301\u8d85\u5e73\u9762\uff0c\u5e76\u4e14\u76f8\u5bf9\u5e94\u5730\uff0c\u5728\u8be5\u70b9\u7684\u6b21\u5fae\u5206\u662f\u4e00\u4e2a\u8fde\u7eed\u7684\u6b21\u68af\u5ea6\u7684\u96c6\u5408\u3002 \u4e00\u4e2a\u5178\u578b\u7684\u4f8b\u5b50\u662f\u7edd\u5bf9\u503c\u51fd\u6570[latex]f(x)=|x|[\/latex]\uff0c\u5b83\u57280\u70b9\u662f\u4e0d\u53ef\u5bfc\u7684\u3002\u4f46\u5728\u8fd9\u4e2a\u70b9\u4e0a\uff0c\u5b83\u53ef\u4ee5\u7531[latex]l(x)=\\alpha x, \\alpha \\in [-1,1][\/latex]\u7ec4\u6210\u7684\u6240\u6709\u76f4\u7ebf\u652f\u6301\u3002\u8fd9\u4e2a\u96c6\u5408\u5373\u8be5\u51fd\u6570\u57280\u70b9\u7684\u6b21\u5fae\u5206\uff0c\u7528[latex]\\partial f(0)[\/latex]\u8868\u793a\u3002 \u51fd\u6570[latex]f(x)=|x|[\/latex]\u7684\u6f14\u793a\uff08\u7c97\u7ebf\u8868\u793a\uff09\u4ee5\u53ca\u5176\u4e2d\u4e24\u4e2a\u652f\u6301\u76f4\u7ebf\uff08\u865a\u7ebf\u8868\u793a\uff09\u3002\u8fd9\u4e24\u6761\u652f\u6301\u76f4\u7ebf\u90fd\u6709\u6b21\u5fae\u5206[latex]\\partial f(0)[\/latex]\u4e2d\u7684\u659c\u7387\u3002\u6ce8\u610f\uff0c\u90a3\u6761\u6c34\u5e73\u7ebf\u4e5f\u662f\u652f\u6301\u8d85\u5e73\u9762\u4e4b\u4e00\uff0c\u8868\u660e[latex]0 \\in \\partial f(0)[\/latex]\u3002\u5e76\u4e14\u56e0\u6b64\u7531\u4e00\u9636\u6761\u4ef6(\u4e0b\u6587\u5b9a\u4e49)\uff0c\u8fd9\u4e2a\u51fd\u6570\u5728\u539f\u70b9\u6709\u4e00\u4e2a\u6781\u5c0f\u503c\u3002 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[162,170],"class_list":["post-2351","post","type-post","status-publish","format-standard","hentry","category-study","tag-162","tag-170"],"_links":{"self":[{"href":"https:\/\/dayandcarrot.space\/index.php?rest_route=\/wp\/v2\/posts\/2351","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dayandcarrot.space\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dayandcarrot.space\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/dayandcarrot.space\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/dayandcarrot.space\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2351"}],"version-history":[{"count":0,"href":"https:\/\/dayandcarrot.space\/index.php?rest_route=\/wp\/v2\/posts\/2351\/revisions"}],"wp:attachment":[{"href":"https:\/\/dayandcarrot.space\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2351"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dayandcarrot.space\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2351"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dayandcarrot.space\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2351"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}