{"id":4772,"date":"2025-03-31T14:33:41","date_gmt":"2025-03-31T05:33:41","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=4772"},"modified":"2025-08-19T16:32:34","modified_gmt":"2025-08-19T07:32:34","slug":"9-nlopt1","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/tutorial\/9-nlopt1\/","title":{"rendered":"9. \u975e\u7dda\u5f62\u6700\u9069\u5316 (1 \u5909\u6570)"},"content":{"rendered":"\n

9.1 \u6982\u8981<\/h3>\n

1 \u5909\u6570\u306e\u975e\u7dda\u5f62\u95a2\u6570 (\u4e00\u822c\u95a2\u6570) \u306e\u6700\u5c0f\u70b9 (\u7b26\u53f7\u3092\u9006\u306b\u3059\u308c\u3070\u6700\u5927\u70b9) \u3092\u6c42\u3081\u308b\u554f\u984c\u3092\u8003\u3048\u307e\u3059.
\n\\[
\nmin f(x)
\n\\]\n\u3053\u3053\u3067, \u5bfe\u8c61\u3068\u3059\u308b\u9818\u57df\u5185\u3067\u306f 1 \u3064\u306e\u6700\u5c0f\u70b9\u3057\u304b\u6301\u305f\u306a\u3044\u3082\u306e\u3068\u3057\u307e\u3059. \u3059\u306a\u308f\u3061, \u76ee\u7684\u3068\u3059\u308b\u6700\u5c0f\u70b9\u306e\u8fd1\u508d\u3060\u3051\u306b\u7740\u76ee\u3059\u308b\u3053\u3068\u306b\u3057\u307e\u3059. \u3053\u308c\u3092\u5c40\u6240\u7684\u6700\u9069\u5316\u3042\u308b\u3044\u306f\u5c40\u6240\u7684\u6700\u5c0f\u70b9\u3092\u6c42\u3081\u308b\u306a\u3069\u3068\u3044\u3044\u307e\u3059.<\/p>\n

9.2 \u975e\u7dda\u5f62\u6700\u9069\u5316\u306e\u89e3\u6cd5 (1 \u5909\u6570)<\/h3>\n

9.2.1 \u9ec4\u91d1\u5206\u5272\u63a2\u7d22\u6cd5<\/h4>\n

\u521d\u671f\u5024\u3068\u3057\u3066 \\(a\\) \u3068 \\(b\\) \u3092\u4e0e\u3048\u307e\u3059. \u95a2\u6570\u5024\u306f \\(f(a)\\) \u3068 \\(f(b)\\) \u3067\u3059. \u6700\u5c0f\u70b9 1 \u3064\u304c\u533a\u9593 \\([a, b]\\) \u306b\u5165\u3063\u3066\u3044\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093.<\/p>\n

\\(r = (\\sqrt{5} – 1)\/2 (\\simeq 0.618)\\) \u306f\u9ec4\u91d1\u6bd4(\u306e\u9006\u6570)\u3068\u547c\u3070\u308c\u308b\u5024\u3067\u3059. \u3053\u308c\u306b\u3064\u3044\u3066 \\(r^2 = 1 – r\\) \u304c\u6210\u308a\u7acb\u3061\u307e\u3059.<\/p>\n

\u533a\u9593\u5e45\\(\\times (1 – r)\\) \u304a\u3088\u3073 \u533a\u9593\u5e45\\(\\times r\\) \u306e\u70b9\u3092 \\(c\\) \u304a\u3088\u3073 \\(d\\) \u3068\u3057\u307e\u3059. \u3059\u306a\u308f\u3061, \\(c = a + (b – a)(1 – r), d = a + (b – a)r\\) \u3068\u3057\u307e\u3059. \u305d\u3057\u3066, \u95a2\u6570\u5024 \\(f(c)\\) \u304a\u3088\u3073 \\(f(d)\\) \u3092\u8a08\u7b97\u3057\u307e\u3059.<\/p>\n

\"\"<\/p>\n

\\(f(c) < f(d)\\) \u306a\u3089\u3070, \\(d\\) \u3092\u65b0\u3057\u3044 \\(b\\) \u3068\u3057\u3066\u533a\u9593\u3092\u7e2e\u5c0f\u3057\u307e\u3059. \u305d\u3046\u3059\u308b\u3068, \\(\u65b0\u533a\u9593\u5e45 = \u65e7\u533a\u9593\u5e45 \\times r\\) \u3068\u306a\u308a\u307e\u3059. \u65b0\u3057\u3044 \\(c\\) \u3068 \\(d\\) \u306f\n\\[\n\\begin{align}\n& \u65b0c: \u65b0\u533a\u9593\u5e45 \\times (1 - r) = \u65e7\u533a\u9593\u5e45 \\times (1 - r)r \\space \u306e\u70b9 \\\\\n& \u65b0d: \u65b0\u533a\u9593\u5e45 \\times r = \u65e7\u533a\u9593\u5e45 \\times r^2 = \u65e7\u533a\u9593\u5e45 \\times (1- r) \\space \u306e\u70b9\n\\end{align}\n\\]\n\u3068\u306a\u308a, \\(\u65b0d = \u65e7c\\) \u3068\u306a\u3063\u3066\u3044\u307e\u3059. \u3059\u306a\u308f\u3061, \u65b0\u533a\u9593\u3067\u306f \\(\u65b0c\\) \u3067\u306e\u95a2\u6570\u5024\u3060\u3051\u3092\u8a08\u7b97\u3059\u308c\u3070\u3088\u3044\u3053\u3068\u306b\u306a\u308a\u307e\u3059.\n\n\\(f(c) > f(d)\\) \u306a\u3089\u3070, \\(c\\) \u3092\u65b0\u3057\u3044 \\(a\\) \u3068\u3057\u3066\u533a\u9593\u3092\u7e2e\u5c0f\u3057, \u540c\u69d8\u306b \\(\u65b0d\\) \u3067\u306e\u95a2\u6570\u5024\u3060\u3051\u3092\u8a08\u7b97\u3059\u308c\u3070\u3088\u304f\u306a\u308a\u307e\u3059.<\/p>\n

\"\"<\/p>\n

\u4ee5\u4e0a\u3092\u7e70\u308a\u8fd4\u3057\u3066\u533a\u9593\u5e45\u304c\u5341\u5206\u306b\u5c0f\u3055\u304f\u306a\u3063\u305f\u3089\u7d42\u4e86\u3057\u307e\u3059.<\/p>\n

\u3053\u306e\u65b9\u6cd5\u306f\u9ec4\u91d1\u5206\u5272\u63a2\u7d22\u6cd5\u3068\u547c\u3070\u308c\u307e\u3059. 1 \u56de\u306e\u7e70\u308a\u8fd4\u3057\u3067\u533a\u9593\u5e45\u306f\u7d04 0.618 \u500d\u306b\u7e2e\u5c0f\u3059\u308b\u306e\u3067, \u4f3c\u305f\u3088\u3046\u306a\u89e3\u6cd5\u3067\u3042\u308b\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u306e\u4e8c\u5206\u6cd5(0.5 \u500d)\u3088\u308a\u5c11\u3057\u53ce\u675f\u304c\u9045\u304f\u306a\u308a\u307e\u3059.<\/p>\n

9.3 XLPack \u3092\u4f7f\u3063\u305f 1 \u5909\u6570\u95a2\u6570\u306e\u975e\u7dda\u5f62\u6700\u9069\u5316\u306e\u89e3\u304d\u65b9<\/h3>\n

VBA \u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Dfmin<\/strong> \u306f\u6700\u5c0f\u70b9\u3092\u542b\u3080\u533a\u9593\u3092\u4e0e\u3048\u308b\u3068\u9ec4\u91d1\u5206\u5272\u63a2\u7d22\u3088\u308a\u901f\u304f\u6700\u5c0f\u70b9\u3092\u898b\u3064\u3051\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u3059. \u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u306e Dfzero<\/strong> \u3068\u540c\u3058\u3088\u3046\u306b\u88dc\u9593\u3068\u7d44\u307f\u5408\u308f\u305b\u308b\u3053\u3068\u306b\u3088\u308a\u9ec4\u91d1\u5206\u5272\u63a2\u7d22\u3088\u308a\u901f\u304f\u53ce\u675f\u3057\u307e\u3059. Dfmin<\/strong> \u306f XLPack \u30bd\u30eb\u30d0\u30fc\u304b\u3089\u4f7f\u3046\u3053\u3068\u3082\u3067\u304d\u307e\u3059.<\/p>\n

\u4f8b\u984c<\/h5>\n

\u6b21\u306e\u95a2\u6570\u306e \\(x = 1\\) \u4ed8\u8fd1\u306e\u6700\u5c0f\u70b9\u3092\u6c42\u3081\u308b.
\n\\[
\nf(x) = x^3 – 2x – 5
\n\\]\n\\(f(x)\\) \u306e\u5f62\u306f\u6b21\u306e\u3068\u304a\u308a\u3067, \u5927\u57df\u7684\u306b\u306f\u6700\u5c0f\u70b9\u306f\u306a\u3044 (\\(-\\infty\\) \u306b\u767a\u6563\u3059\u308b) \u3082\u306e\u306e, \\(x = 1\\) \u4ed8\u8fd1\u306b\u5c40\u6240\u7684\u306a\u6700\u5c0f\u70b9\u304c\u3042\u308b\u306e\u304c\u308f\u304b\u308a\u307e\u3059.<\/p>\n

\"\"<\/p>\n

9.3.1 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (1)<\/h4>\n

Dfmin<\/strong> \u3092\u4f7f\u3063\u305f VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n

\n
Function F(X As Double) As Double\r\n    F = X ^ 3 - 2 * X - 5\r\nEnd Function\r\n\r\nSub Start()\r\n    Dim Ax As Double, Bx As Double, Tol As Double, X As Double\r\n    '--- Input data\r\n    Ax = 0\r\n    Bx = 1.5\r\n    Tol = 0.000000000000001 '1e-15\r\n    '--- Compute min. point of equation\r\n    Call Dfmin(Ax, Bx, AddressOf F, Tol, X)\r\n    '--- Output min. point\r\n    MsgBox X\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
Dfmin<\/strong> \u306f, \u4e0e\u3048\u3089\u308c\u305f\u533a\u9593 \\([Ax, Bx]\\) \u3067\u5c40\u6240\u7684\u306a\u6700\u5c0f\u70b9\u304c 1 \u3064\u631f\u307e\u308c\u3066\u3044\u308b\u3082\u306e\u3068\u3057\u3066, \u305d\u306e\u6700\u5c0f\u70b9\u3092\u6c42\u3081\u307e\u3059. \u3053\u306e\u5834\u5408, \u6700\u5c0f\u70b9\u306f 1 \u4ed8\u8fd1\u306b\u3042\u308b\u306e\u3067, \\(Ax = 0, Bx = 1.5\\) \u3068\u3057\u307e\u3057\u305f.<\/p>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068, \\(x = 0.81650\\) \u304c\u6c42\u3081\u3089\u308c\u307e\u3059.<\/p>\n
\"\"<\/p>\n
9.3.2 VBA \u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9 (2)<\/h4>\n
\u30ea\u30d0\u30fc\u30b9\u30b3\u30df\u30e5\u30cb\u30b1\u30fc\u30b7\u30e7\u30f3\u7248 (RCI) \u306e Dfmin_r<\/strong> \u3092\u4f7f\u3063\u305f\u30d7\u30ed\u30b0\u30e9\u30e0\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n
\n
Sub Start()\r\n    Dim Ax As Double, Bx As Double, Tol As Double, X As Double\r\n    Dim XX As Double, YY As Double, IRev As Long\r\n    '--- Input data\r\n    Ax = 0\r\n    Bx = 1.5\r\n    Tol = 0.000000000000001 '1e-15\r\n    '--- Compute min. point of equation\r\n    IRev = 0\r\n    Do\r\n        Call Dfmin_r(Ax, Bx, Tol, XX, YY, IRev)\r\n        If IRev <> 0 Then YY = XX ^ 3 - 2 * XX - 5\r\n    Loop While IRev <> 0\r\n    X = XX\r\n    '--- Output min. point\r\n    MsgBox X\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u76ee\u7684\u95a2\u6570\u3092\u5916\u90e8\u95a2\u6570\u3068\u3057\u3066\u4e0e\u3048\u308b\u306e\u3067\u306f\u306a\u304f, IRev = 1 \u306e\u3068\u304d\u306b XX \u306e\u5024\u3092\u4f7f\u3063\u3066\u95a2\u6570\u5024\u3092\u8a08\u7b97\u3057 YY \u306b\u5165\u308c\u3066\u518d\u5ea6 Dfmin_r<\/strong> \u3092\u547c\u3073\u51fa\u3057\u307e\u3059. RCI \u306e\u8a73\u7d30\u306b\u3064\u3044\u3066\u306f \u3053\u3061\u3089<\/a> \u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044.<\/p>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068\u4e0a\u3068\u540c\u3058\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3059.<\/p>\n
9.3.3 \u30bd\u30eb\u30d0\u30fc\u3092\u4f7f\u7528\u3057\u305f\u89e3\u304d\u65b9<\/h4>\n
XLPack\u30bd\u30eb\u30d0\u30fc\u30a2\u30c9\u30a4\u30f3\u306e\u300c\uff11\u5909\u6570\u975e\u7dda\u5f62\u6700\u9069\u5316\u300d\u3092\u4f7f\u3063\u3066\u89e3\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059. B7\u30bb\u30eb\u306b\u6570\u5f0f (=B6^3-2*B6-5) \u304c\u5165\u529b\u3055\u308c\u3066\u3044\u307e\u3059.<\/p>\n
\"\"<\/p>\n
\u30bd\u30eb\u30d0\u30fc\u306b\u3064\u3044\u3066\u306f \u3053\u3061\u3089<\/a> \u3082\u53c2\u7167\u304f\u3060\u3055\u3044.<\/p>\n","protected":false},"excerpt":{"rendered":"
\u975e\u7dda\u5f62\u6700\u9069\u5316(1 \u5909\u6570)\u306e\u89e3\u6cd5\u3068XLPack\u3092\u4f7f\u3063\u305f\u89e3\u304d\u65b9 <\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[10],"tags":[],"class_list":["post-4772","post","type-post","status-publish","format-standard","hentry","category-tutorial"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4772","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/comments?post=4772"}],"version-history":[{"count":5,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4772\/revisions"}],"predecessor-version":[{"id":5132,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4772\/revisions\/5132"}],"wp:attachment":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/media?parent=4772"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/categories?post=4772"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/tags?post=4772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}