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全双対整数性 - 版の履歴
2026-04-16T17:00:25Z
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Orsjwiki: "全双対整数性" を保護しました。 [edit=sysop:move=sysop]
2007-07-20T03:02:55Z
<p>"全双対整数性" を保護しました。 [edit=sysop:move=sysop]</p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
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<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">← 古い版</td>
<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">2007年7月20日 (金) 03:02時点における版</td>
</tr><tr><td colspan="2" class="diff-notice" lang="ja"><div class="mw-diff-empty">(相違点なし)</div>
</td></tr></table>
Orsjwiki
https://orsj-ml.org/orwiki/wiki/index.php?title=%E5%85%A8%E5%8F%8C%E5%AF%BE%E6%95%B4%E6%95%B0%E6%80%A7&diff=5323&oldid=prev
2007年7月17日 (火) 05:52に122.17.2.240による
2007-07-17T05:52:34Z
<p></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
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<col class="diff-marker" />
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← 古い版</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2007年7月17日 (火) 05:52時点における版</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >1行目:</td>
<td colspan="2" class="diff-lineno">1行目:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''【ぜんそうついせいすうせい (totally dual integrality (TDI))】'''</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''【ぜんそうついせいすうせい (totally dual integrality (TDI))】'''</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>線形不等式システム <math>\<del class="diffchange diffchange-inline">mathbf</del>{A} \<del class="diffchange diffchange-inline">mathbf</del>{x} \leq \<del class="diffchange diffchange-inline">mathbf</del>{b} \,</math> が線形計画問題 <math> \max \{\<del class="diffchange diffchange-inline">mathbf</del>{c} \<del class="diffchange diffchange-inline">mathbf</del>{x} \mid \<del class="diffchange diffchange-inline">mathbf</del>{A} \<del class="diffchange diffchange-inline">mathbf</del>{x} \leq \<del class="diffchange diffchange-inline">mathbf</del>{b} \} \,</math> が有界であるような任意の整数ベクトル <math>\<del class="diffchange diffchange-inline">mathbf</del>{c} \,</math> に対して, その双対問題 <math> \min\{ \<del class="diffchange diffchange-inline">mathbf</del>{b} \<del class="diffchange diffchange-inline">mathbf</del>{y} \mid \<del class="diffchange diffchange-inline">mathbf</del>{y} \<del class="diffchange diffchange-inline">mathbf</del>{A} = \<del class="diffchange diffchange-inline">mathbf</del>{c}, \<del class="diffchange diffchange-inline">mathbf</del>{y} \geq \<del class="diffchange diffchange-inline">mathbf</del>{0} \} \,</math>が, 整数の最適解 <math>\<del class="diffchange diffchange-inline">mathbf</del>{y}^* \,</math> をもつならば, 全双対整数的 (totally dual integral, TDI) であるという.</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>線形不等式システム <math>\<ins class="diffchange diffchange-inline">boldsymbol</ins>{A} \<ins class="diffchange diffchange-inline">boldsymbol</ins>{x} \leq \<ins class="diffchange diffchange-inline">boldsymbol</ins>{b} \,</math> が線形計画問題 <math> \max \{\<ins class="diffchange diffchange-inline">boldsymbol</ins>{c} \<ins class="diffchange diffchange-inline">boldsymbol</ins>{x} \mid \<ins class="diffchange diffchange-inline">boldsymbol</ins>{A} \<ins class="diffchange diffchange-inline">boldsymbol</ins>{x} \leq \<ins class="diffchange diffchange-inline">boldsymbol</ins>{b} \} \,</math> が有界であるような任意の整数ベクトル <math>\<ins class="diffchange diffchange-inline">boldsymbol</ins>{c} \,</math> に対して, その双対問題 <math> \min\{ \<ins class="diffchange diffchange-inline">boldsymbol</ins>{b} \<ins class="diffchange diffchange-inline">boldsymbol</ins>{y} \mid \<ins class="diffchange diffchange-inline">boldsymbol</ins>{y} \<ins class="diffchange diffchange-inline">boldsymbol</ins>{A} = \<ins class="diffchange diffchange-inline">boldsymbol</ins>{c}, \<ins class="diffchange diffchange-inline">boldsymbol</ins>{y} \geq \<ins class="diffchange diffchange-inline">boldsymbol</ins>{0} \} \,</math>が, 整数の最適解 <math>\<ins class="diffchange diffchange-inline">boldsymbol</ins>{y}^* \,</math> をもつならば, 全双対整数的 (totally dual integral, TDI) であるという.</div></td></tr>
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122.17.2.240
https://orsj-ml.org/orwiki/wiki/index.php?title=%E5%85%A8%E5%8F%8C%E5%AF%BE%E6%95%B4%E6%95%B0%E6%80%A7&diff=4669&oldid=prev
2007年7月13日 (金) 16:49に124.144.188.143による
2007-07-13T16:49:44Z
<p></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="ja">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← 古い版</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2007年7月13日 (金) 16:49時点における版</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >1行目:</td>
<td colspan="2" class="diff-lineno">1行目:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''【ぜんそうついせいすうせい (totally dual integrality (TDI))】'''</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''【ぜんそうついせいすうせい (totally dual integrality (TDI))】'''</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>線形不等式システム <del class="diffchange diffchange-inline">$</del>\<del class="diffchange diffchange-inline">mbox</del>{\<del class="diffchange diffchange-inline">boldmath $A </del>x<del class="diffchange diffchange-inline">$</del>} \leq \<del class="diffchange diffchange-inline">mbox</del>{<del class="diffchange diffchange-inline">\boldmath $</del>b<del class="diffchange diffchange-inline">$</del>}<del class="diffchange diffchange-inline">$ </del>が線形計画問題 <del class="diffchange diffchange-inline">$ </del>\max\{\<del class="diffchange diffchange-inline">mbox</del>{\<del class="diffchange diffchange-inline">boldmath $c </del>x<del class="diffchange diffchange-inline">$</del>} \mid \<del class="diffchange diffchange-inline">mbox</del>{\<del class="diffchange diffchange-inline">boldmath $A </del>x<del class="diffchange diffchange-inline">$</del>} \leq \<del class="diffchange diffchange-inline">mbox</del>{<del class="diffchange diffchange-inline">\boldmath $</del>b<del class="diffchange diffchange-inline">$</del>}\}<del class="diffchange diffchange-inline">$ </del>が有界であるような任意の整数ベクトル {<del class="diffchange diffchange-inline">\boldmath $</del>c<del class="diffchange diffchange-inline">$</del>} に対して, その双対問題 <del class="diffchange diffchange-inline">$ </del>\min\{\<del class="diffchange diffchange-inline">mbox</del>{\<del class="diffchange diffchange-inline">boldmath $b </del>y<del class="diffchange diffchange-inline">$</del>} \mid \<del class="diffchange diffchange-inline">mbox</del>{\<del class="diffchange diffchange-inline">boldmath $y </del>A<del class="diffchange diffchange-inline">$</del>} = \<del class="diffchange diffchange-inline">mbox</del>{<del class="diffchange diffchange-inline">\boldmath $</del>c<del class="diffchange diffchange-inline">$</del>}, \<del class="diffchange diffchange-inline">mbox</del>{<del class="diffchange diffchange-inline">\boldmath $</del>y<del class="diffchange diffchange-inline">$</del>} \geq \<del class="diffchange diffchange-inline">mbox</del>{<del class="diffchange diffchange-inline">\boldmath $</del>0<del class="diffchange diffchange-inline">$</del>}\} <del class="diffchange diffchange-inline">$</del>が, 整数の最適解 <del class="diffchange diffchange-inline">$</del>\<del class="diffchange diffchange-inline">mbox</del>{<del class="diffchange diffchange-inline">\boldmath $</del>y<del class="diffchange diffchange-inline">$</del>}^*<del class="diffchange diffchange-inline">$ </del>をもつならば, 全双対整数的 (totally dual integral, TDI) であるという.</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>線形不等式システム <ins class="diffchange diffchange-inline"><math></ins>\<ins class="diffchange diffchange-inline">mathbf</ins>{<ins class="diffchange diffchange-inline">A} </ins>\<ins class="diffchange diffchange-inline">mathbf{</ins>x} \leq \<ins class="diffchange diffchange-inline">mathbf</ins>{b} <ins class="diffchange diffchange-inline">\,</math> </ins>が線形計画問題 <ins class="diffchange diffchange-inline"><math> </ins>\max \{\<ins class="diffchange diffchange-inline">mathbf</ins>{<ins class="diffchange diffchange-inline">c} </ins>\<ins class="diffchange diffchange-inline">mathbf{</ins>x} \mid \<ins class="diffchange diffchange-inline">mathbf</ins>{<ins class="diffchange diffchange-inline">A} </ins>\<ins class="diffchange diffchange-inline">mathbf{</ins>x} \leq \<ins class="diffchange diffchange-inline">mathbf</ins>{b} \} <ins class="diffchange diffchange-inline">\,</math> </ins>が有界であるような任意の整数ベクトル <ins class="diffchange diffchange-inline"><math>\mathbf</ins>{c} <ins class="diffchange diffchange-inline">\,</math> </ins>に対して, その双対問題 <ins class="diffchange diffchange-inline"><math> </ins>\min\{ \<ins class="diffchange diffchange-inline">mathbf</ins>{<ins class="diffchange diffchange-inline">b} </ins>\<ins class="diffchange diffchange-inline">mathbf{</ins>y} \mid \<ins class="diffchange diffchange-inline">mathbf</ins>{<ins class="diffchange diffchange-inline">y} </ins>\<ins class="diffchange diffchange-inline">mathbf{</ins>A} = \<ins class="diffchange diffchange-inline">mathbf</ins>{c}, \<ins class="diffchange diffchange-inline">mathbf</ins>{y} \geq \<ins class="diffchange diffchange-inline">mathbf</ins>{0} \} <ins class="diffchange diffchange-inline">\,</math></ins>が, 整数の最適解 <ins class="diffchange diffchange-inline"><math></ins>\<ins class="diffchange diffchange-inline">mathbf</ins>{y}^* <ins class="diffchange diffchange-inline">\,</math> </ins>をもつならば, 全双対整数的 (totally dual integral, TDI) であるという.</div></td></tr>
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124.144.188.143
https://orsj-ml.org/orwiki/wiki/index.php?title=%E5%85%A8%E5%8F%8C%E5%AF%BE%E6%95%B4%E6%95%B0%E6%80%A7&diff=3724&oldid=prev
122.17.2.240: 新しいページ: ''''【ぜんそうついせいすうせい (totally dual integrality (TDI))】''' 線形不等式システム $\mbox{\boldmath $A x$} \leq \mbox{\boldmath $b$}$ が線形計...'
2007-07-12T15:54:11Z
<p>新しいページ: ''''【ぜんそうついせいすうせい (totally dual integrality (TDI))】''' 線形不等式システム $\mbox{\boldmath $A x$} \leq \mbox{\boldmath $b$}$ が線形計...'</p>
<p><b>新規ページ</b></p><div>'''【ぜんそうついせいすうせい (totally dual integrality (TDI))】'''<br />
<br />
線形不等式システム $\mbox{\boldmath $A x$} \leq \mbox{\boldmath $b$}$ が線形計画問題 $ \max\{\mbox{\boldmath $c x$} \mid \mbox{\boldmath $A x$} \leq \mbox{\boldmath $b$}\}$ が有界であるような任意の整数ベクトル {\boldmath $c$} に対して, その双対問題 $ \min\{\mbox{\boldmath $b y$} \mid \mbox{\boldmath $y A$} = \mbox{\boldmath $c$}, \mbox{\boldmath $y$} \geq \mbox{\boldmath $0$}\} $が, 整数の最適解 $\mbox{\boldmath $y$}^*$ をもつならば, 全双対整数的 (totally dual integral, TDI) であるという.</div>
122.17.2.240