\u62c9\u683c\u6717\u65e5\u4e58\u5b50(Lagrange Multiplier)\u7528\u5728\u542b\u6709\u7b49\u5f0f\u7ea6\u675f\u7684\u6700\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u5982\u76ee\u6807\u51fd\u6570\u662f$$f(x,y,…)$$\uff0c\u7ea6\u675f\u6761\u4ef6$g(x,y,…) = 0$\uff0c\u5f15\u5165\u62c9\u683c\u6717\u65e5\u4e58\u5b50$\\lambda$\u7528\u6765\u534f\u52a9\u6c42\u89e3\u4f7f$f$\u6700\u5927\u6216\u6700\u5c0f\u7684\u89e3\u3002<\/p>\n\n\n\n
\u5229\u7528\u62c9\u683c\u6717\u65e5\u4e58\u5b50\u5c06\u76ee\u6807\u51fd\u6570\u548c\u7b49\u5f0f\u7ea6\u675f\u6761\u4ef6\u7ec4\u5408\u8d77\u6765\u79f0\u4e3a\u62c9\u683c\u6717\u65e5\u65b9\u7a0b(Lagrange Function(L))<\/p>\n\n\n\n
$$L(x, y, \\lambda) = f(x, y) – \\lambda \\cdot g(x,y)$$<\/p>\n\n\n\n
\u6c42\u89e3\u4e0a\u8ff0\u95ee\u9898\uff0c\u9700\u8981\u5206\u522b\u5bf9\u51e0\u4e2a\u672a\u77e5\u53d8\u91cf\u6c42\u4e00\u9636\u5bfc\u6570\uff0c\u4ee4\u4ed6\u4eec\u7b49\u4e8e0\u53ef\u4ee5\u627e\u5230\u9a7b\u70b9(critical points):<\/p>\n\n\n\n
$$\\frac{\\partial L}{\\partial x} = 0, \\frac{\\partial L}{\\partial y} = 0, \\frac{\\partial L}{\\partial \\lambda} = 0$$<\/p>\n\n\n\n
\u6839\u636e\u4e0a\u9762\u8ba1\u7b97\u51fa\u6765\u7684\u7ed3\u679c\u627e\u5230\u9700\u8981\u7684\u89e3\u5c31\u53ef\u4ee5\u4e86\u3002<\/p>\n\n\n\n
\u6765\u4e2a\u4f8b\u5b50\u5427\uff0c\u7ea6\u675f$x+y=1$\u7684\u65f6\u5019\u6c42\u89e3\u4e0b\u76ee\u6807\u51fd\u6570\u7684\u6700\u5c0f\u503c<\/p>\n\n\n\n
$$f(x,y) = x^2 + y^2$$<\/p>\n\n\n\n
1.\u7ec4\u6210\u62c9\u683c\u6717\u65e5\u65b9\u7a0b<\/p>\n\n\n\n
$$L(x,y,\\lambda) = x^2 + y^2 – \\lambda (x + y – 1)$$<\/p>\n\n\n\n
2.\u8ba1\u7b97\u504f\u5bfc<\/p>\n\n\n\n
$$\\frac{\\partial L}{\\partial x} = 2x-\\lambda, \\frac{\\partial L}{\\partial y} = 2y-\\lambda, \\frac{\\partial L}{\\partial \\lambda} = -(x + y – 1)$$<\/p>\n\n\n\n
3.\u4e8e\u662f<\/p>\n\n\n\n
$$2x – \\lambda = 0$$<\/p>\n\n\n\n
$$2y – \\lambda = 0$$<\/p>\n\n\n\n
$$x + y = 1$$<\/p>\n\n\n\n
\u4e0a\u4e70\u4e24\u4e2a\u7b49\u5f0f\u544a\u8bc9\u6211\u4eec$2x=2y$\uff0c\u6240\u4ee5$x=y$\uff0c\u5e26\u5165\u7b2c\u4e09\u4e2a\u7b49\u5f0f\u5f88\u5bb9\u6613\u7b97\u51fa\u6765$x=\\frac{1}{2},y=\\frac{1}{2}$\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u62c9\u683c\u6717\u65e5\u4e58\u5b50(Lagrange Multiplier)\u7528\u5728\u542b\u6709\u7b49\u5f0f\u7ea6\u675f\u7684\u6700\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u5982\u76ee\u6807\u51fd\u6570\u662f$$f(x,y,…)$$\uff0c\u7ea6\u675f\u6761\u4ef6$g(x,y,…) = 0$\uff0c\u5f15\u5165\u62c9\u683c\u6717\u65e5\u4e58\u5b50$\\lambda$\u7528\u6765\u534f\u52a9\u6c42\u89e3\u4f7f$f$\u6700\u5927\u6216\u6700\u5c0f\u7684\u89e3\u3002<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-649","post","type-post","status-publish","format-standard","hentry","category-matchematics"],"_links":{"self":[{"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/posts\/649","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=649"}],"version-history":[{"count":3,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/posts\/649\/revisions"}],"predecessor-version":[{"id":652,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=\/wp\/v2\/posts\/649\/revisions\/652"}],"wp:attachment":[{"href":"https:\/\/tensorzen.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=649"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=649"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tensorzen.blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=649"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}