{"id":1716,"date":"2026-06-07T01:40:47","date_gmt":"2026-06-06T17:40:47","guid":{"rendered":"https:\/\/okb385.org.cn\/?p=1716"},"modified":"2026-08-02T19:09:20","modified_gmt":"2026-08-02T11:09:20","slug":"transformation-between-the-primal-and-dual-problems-in-linear-programming","status":"publish","type":"post","link":"https:\/\/okb385.org.cn\/index.php\/2026\/06\/07\/transformation-between-the-primal-and-dual-problems-in-linear-programming\/","title":{"rendered":"\u7ebf\u6027\u89c4\u5212\u539f\u95ee\u9898\u4e0e\u5bf9\u5076\u95ee\u9898\u7684\u8f6c\u6362"},"content":{"rendered":"\n\n\n\n\n
<\/td>\u539f\u95ee\u9898<\/td>\u5bf9\u5076\u95ee\u9898<\/td><\/tr>
\u53d8\u91cf<\/td>$x_1,\\cdots,x_n$<\/td>$y_1,\\cdots,y_m$<\/td><\/tr>
\u77e9\u9635<\/td>$A$<\/td>$A^T$<\/td><\/tr>
\u4e0d\u7b49\u5f0f\u53f3\u65b9\u5e38\u6570<\/td>$b$<\/td>$c$<\/td><\/tr>
\u76ee\u6807\u51fd\u6570<\/td>$\\max c^Tx$<\/td>$\\min b^T y$<\/td><\/tr>
\u7ea6\u675f<\/td>\u7b2c$i$\u6761\u7ea6\u675f\u662f$\\le$<\/td>$y_i\\ge0$<\/td><\/tr>
<\/td>$\\ge$<\/td>$y_i\\le0$<\/td><\/tr>
<\/td>$=$<\/td>$y_i\\in\\F$<\/td><\/tr>
<\/td>$x_j\\ge0$<\/td>\u7b2c$j$\u6761\u7ea6\u675f\u662f$\\ge$<\/td><\/tr>
<\/td>$x_j\\le0$<\/td>$\\le$<\/td><\/tr>
<\/td>$x_j\\in\\F$<\/td>$=$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

\u9996\u5148\u89e3\u91ca\u8868\u683c\u4e2d\u53cd\u76f4\u89c9\u7684\u7ea6\u675f\u7b26\u53f7\u5bf9\u5e94\u53d8\u91cf\u53d6\u503c\u7b26\u53f7\u7684\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u539f\u95ee\u9898$\\le$\u7684\u7ea6\u675f\u5bf9\u5e94\u5bf9\u5076\u53d8\u91cf$\\ge0$\u7684\u53d6\u503c\uff0c\u800c\u5bf9\u5076\u95ee\u9898$\\ge$\u7684\u7ea6\u675f\u540c\u6837\u5bf9\u5e94\u539f\u53d8\u91cf$\\ge0$\u7684\u53d6\u503c\u3002<\/p>\n\n\n\n

\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u6c42\u6700\u5927\u503c\u7684\u7ebf\u6027\u89c4\u5212\u539f\u95ee\u9898\n\\begin{align*}\n\\max&~c^T x\\\\Ax&\\le b\\\\x&\\ge0\n\\end{align*}\n\u5bf9\u5076\u95ee\u9898\u9700\u8981\u901a\u8fc7\u5bf9\u7ea6\u675f\u6761\u4ef6\u8fdb\u884c\u7ebf\u6027\u7ec4\u5408\uff0c\u4ee5\u627e\u5230\u4e00\u4e2a\u539f\u95ee\u9898\u7684\u4e0a\u754c\u3002\u5373\uff0c\u5728$Ax\\le b$\u4e24\u8fb9\u540c\u4e58\u53d8\u91cf$y^T$\u5f97\u5230\n\\begin{align*}\ny^TAx&\\le y^Tb\\\\\n&\\Longleftrightarrow\\\\\n(A^Ty)^T x&\\le b^Ty\n\\end{align*}\n\n\n\n

\u9700\u4fdd\u6301$\\le$\u65b9\u5411\uff0c\u56e0\u6b64$y\\ge0$\u3002\u82e5\u9009\u53d6$y$\u4f7f\u5f97$c\\le A^T y$\uff0c\u6211\u4eec\u5c31\u5f97\u5230$c^T x$\u7684\u4e0a\u754c$b^T y$\u3002\u56e0\u6b64\u5bf9\u5076\u95ee\u9898\u5c31\u662f\u5728$y\\ge0$\u4ee5\u53ca$A^T y\\ge c$\u7684\u7ea6\u675f\u4e0b\uff0c\u6700\u5c0f\u5316$c^Tx$\u7684\u4e0a\u754c$b^Ty$\u3002<\/p>\n\n\n\n

\u540c\u7406\uff0c\u82e5\u5bf9\u5076\u95ee\u9898\u662f\n\\begin{align*}\n\\min&~b^T y\\\\A^T y&\\ge c\\\\y&\\ge0\n\\end{align*}\n\n\u539f\u95ee\u9898\u9700\u8981\u901a\u8fc7\u5bf9\u7ea6\u675f\u6761\u4ef6\u8fdb\u884c\u7ebf\u6027\u7ec4\u5408\uff0c\u4ee5\u627e\u5230\u4e00\u4e2a\u5bf9\u5076\u95ee\u9898\u7684\u4e0b\u754c\u3002\u5373\uff0c\u5728$A^Ty\\ge c$\u4e24\u8fb9\u540c\u4e58\u53d8\u91cf$x^T$\u5f97\u5230\n\\begin{align*}\nx^TA^Ty&\\ge x^Tc\\\\\n&\\Longleftrightarrow\\\\\n(Ax)^T y&\\ge c^T x\n\\end{align*}\n\n\n

\u9700\u4fdd\u6301$\\ge$\u65b9\u5411\uff0c\u56e0\u6b64$x\\ge0$\u3002\u82e5\u9009\u53d6$x$\u4f7f\u5f97$b\\ge A x$\uff0c\u6211\u4eec\u5c31\u5f97\u5230$b^T y$\u7684\u4e0b\u754c$c^T x$\u3002\u56e0\u6b64\u539f\u95ee\u9898\u5c31\u662f\u5728$x\\ge0$\u4ee5\u53ca$A x\\le b$\u7684\u7ea6\u675f\u4e0b\uff0c\u6700\u5927\u5316$b^T y$\u7684\u4e0b\u754c$c^T x$\u3002<\/p>\n\n\n

\u6211\u4eec\u8003\u8651\u5bf9\u4e8e\u6c42\u6700\u5927\u503c\u7684\u7ebf\u6027\u89c4\u5212\u6765\u8bf4\uff0c\u4ec0\u4e48\u662f\u201c\u6b63\u5e38\u201d\u7684\u3002\u5373\uff0c\u7ea6\u675f\u4e3a$\\le$\u4ee5\u9632\u6b62\u53d8\u91cf\u65e0\u9650\u5236\u589e\u5927\uff0c\u4e14\u53d6\u503c$\\ge0$\u3002\u5bf9\u4e8e\u6c42\u6700\u5c0f\u503c\u7684\u7ebf\u6027\u89c4\u5212\uff0c\u7ea6\u675f\u4e3a$\\ge$\u4ee5\u9632\u6b62\u53d8\u91cf\u65e0\u9650\u5236\u51cf\u5c0f\uff0c\u53d8\u91cf\u540c\u6837\u53d6\u503c$\\ge0$\u3002<\/p>\n\n\n

\u5728\u8f6c\u6362\u5bf9\u5076\u95ee\u9898\u65f6\uff0c\u201c\u6b63\u5e38\u201d\u7684\u539f\u95ee\u9898\u7ea6\u675f\uff0c\u5bf9\u5e94\u201c\u6b63\u5e38\u201d\u7684\u5bf9\u5076\u95ee\u9898\u53d8\u91cf\uff0c\u201c\u53cd\u5e38\u201d\u7684\u539f\u95ee\u9898\u7ea6\u675f\uff0c\u5bf9\u5e94\u201d\u53cd\u5e38\u201c\u7684\u5bf9\u5076\u95ee\u9898\u53d8\u91cf\uff0c\u4e25\u683c\u7684\u539f\u95ee\u9898\u7ea6\u675f\uff08\u53d6\u7b49\uff09\uff0c\u5bf9\u5e94\u5bf9\u5076\u95ee\u9898\u7684\u81ea\u7531\u53d8\u91cf\uff08$\\in\\F$\uff09\uff1a<\/p>\n\n\n

\u539f\u95ee\u9898\u6027\u8d28<\/td>\u5bf9\u5076\u95ee\u9898\u6027\u8d28<\/td><\/tr>
$\\max$\u539f\u95ee\u9898\u6b63\u5e38\u7ea6\u675f<\/td>$\\min$\u5bf9\u5076\u95ee\u9898\u6b63\u5e38\u53d8\u91cf<\/td><\/tr>
$\\max$\u539f\u95ee\u9898\u53cd\u5e38\u7ea6\u675f<\/td>$\\min$\u5bf9\u5076\u95ee\u9898\u53cd\u5e38\u53d8\u91cf<\/td><\/tr>
$\\max$\u539f\u95ee\u9898\u4e25\u683c\u7ea6\u675f<\/td>$\\min$\u5bf9\u5076\u95ee\u9898\u81ea\u7531\u53d8\u91cf<\/td><\/tr>
$\\max$\u539f\u95ee\u9898\u6b63\u5e38\u53d8\u91cf<\/td>$\\min$\u5bf9\u5076\u95ee\u9898\u6b63\u5e38\u7ea6\u675f<\/td><\/tr>
$\\max$\u539f\u95ee\u9898\u53cd\u5e38\u53d8\u91cf<\/td>$\\min$\u5bf9\u5076\u95ee\u9898\u53cd\u5e38\u7ea6\u675f<\/td><\/tr>
$\\max$\u539f\u95ee\u9898\u81ea\u7531\u53d8\u91cf<\/td>$\\min$\u5bf9\u5076\u95ee\u9898\u4e25\u683c\u7ea6\u675f<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n

\u7528\u7f51\u7edc\u6700\u5927\u6d41\u95ee\u9898\u7684\u6309\u8def\u5f84\u5efa\u6a21\u7684\u7ebf\u6027\u89c4\u5212\u4e3e\u4f8b\u3002\u8bbe$G=(V,E)$\u662f\u4e00\u4e2a\u6709\u5411\u56fe\uff0c\u8fb9\u5bb9\u91cf$u:E\\to \\F_{+}$\uff0c$s$\u548c$t$\u5206\u522b\u662f\u6e90\u70b9\u548c\u6c47\u70b9\u3002\u8bbe<\/p>\n\n\n

\\[\n\\mathcal{P}:=\\{P\\subseteq E\\mid P\\text{\u662f\u4e00\u6761$s-t-$\u8def\u5f84}\\}\n\\]<\/p>\n\n\n

\u5219\u6700\u5927\u6d41\u95ee\u9898\u7684\u7ebf\u6027\u89c4\u5212\u6a21\u578b\u4e3a\uff1a<\/p>\n\n\n

\\begin{array}{rl} \\max & \\displaystyle \\sum_{P\\in\\mathcal P} y_P\\\\ \\text{s.t.} & \\displaystyle \\sum_{P\\in\\mathcal P:\\, e\\in P} y_P \\le u(e) \\qquad \\forall e\\in E,\\\\ & y_P\\ge 0 \\qquad \\forall P\\in\\mathcal P. \\end{array}<\/p>\n\n\n

\u8fd0\u7528\u8f6c\u6362\u6cd5\u5219\uff1a\u6bcf\u6761\u8def\u5f84\u6709\u4e00\u4e2a\u201c\u6b63\u5e38\u201d\u53d8\u91cf$y_P$\u4ee3\u8868\u8def\u5f84\u4e0a\u7684\u6d41\u91cf\uff0c\u6bcf\u6761\u8fb9\u6709\u4e00\u6761\u6b63\u5e38\u201c\u7ea6\u675f\u201d\u4fdd\u8bc1\u7ecf\u8fc7\u8fd9\u6761\u8fb9\u7684\u6d41\u91cf\u4e0d\u8d85\u8fc7\u5bb9\u91cf\u3002\u5bf9\u6bcf\u6761\u7ea6\u675f\uff08\u8fb9\uff09\u5f15\u5165\u201c\u6b63\u5e38\u201d\u5bf9\u5076\u53d8\u91cf$x_e$\uff0c\u7ea6\u675f\u7b26\u53f7\u4e3a$\\ge$\uff0c\u53d6\u503c$\\ge0$\u3002\u5bf9\u5076\u95ee\u9898\u6709\u539f\u95ee\u9898\u53d8\u91cf\u6570\u91cf\u6761\u7ea6\u675f\uff0c\u53f3\u8fb9\u5e38\u6570\u662f\u539f\u76ee\u6807\u51fd\u6570\u5185\u5bf9\u5e94\u539f\u53d8\u91cf\u7684\u7cfb\u6570$1$\uff0c\u5de6\u8fb9$e$\u7684\u7ea6\u675f\u7684\u5bf9\u5076\u53d8\u91cf$x_e$\u7684\u7cfb\u6570\u662f\u539f\u53d8\u91cf\u5728$e$\u7684\u7ea6\u675f\u4e2d\u7684\u7cfb\u6570$1$\u3002\u4e8e\u662f\u5f97\u5230\u5176\u5bf9\u5076\u95ee\u9898\u7684\u7ebf\u6027\u89c4\u5212\u6a21\u578b\uff1a<\/p>\n\n\n

\\begin{array}{rl}
\\min & \\displaystyle \\sum_{e\\in E} u(e)x_e\\\\
\\text{s.t.}
& \\displaystyle \\sum_{e\\in P}x_e\\ge 1
\\qquad \\forall P\\in\\mathcal P,\\\\
& x_e\\ge 0
\\qquad \\forall e\\in E.
\\end{array}<\/p>\n\n\n

$x_e=1$\u4ee3\u8868\u5272\u6389\u8fd9\u6761\u8fb9\u3002\u663e\u7136\u8fd9\u5c31\u662f\u6700\u5927\u6d41\u95ee\u9898\u7684\u5bf9\u5076\u95ee\u9898\u2500\u2500\u6700\u5c0f\u5272\u95ee\u9898\u3002<\/p>\n\n\n

\u6ce8\uff1a\u5b9e\u9645\u4e0a\u8fd9\u4e2a\u57fa\u4e8e\u8def\u5f84\u7684\u6a21\u578b\u662f\u6781\u5dee\u7684\uff0c\u56e0\u4e3a\u8def\u5f84\u6570\u91cf\u53ef\u4ee5\u8fbe\u5230\u6307\u6570\u7ea7\u522b\u3002\u4e00\u822c\u6700\u5927\u6d41\u95ee\u9898\u7684\u7ebf\u6027\u89c4\u5212\u6a21\u578b\u57fa\u4e8e\u8282\u70b9\u7684\u6d41\u5e73\u8861\u3002<\/p>\n\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_import_markdown_pro_load_document_selector":0,"_import_markdown_pro_submit_text_textarea":"","footnotes":""},"categories":[3,78,79],"tags":[71,80,74],"class_list":["post-1716","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-discrete-mathematics","category-linear-programming","tag-71","tag-80","tag-74"],"_links":{"self":[{"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/posts\/1716","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1716"}],"version-history":[{"count":59,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/posts\/1716\/revisions"}],"predecessor-version":[{"id":1880,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/posts\/1716\/revisions\/1880"}],"wp:attachment":[{"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1716"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1716"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}