線形計画法とカーマーカー法〜プログラマのための数学勉強会#3〜

by Shintaro Kaneko

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ઢܗܭը๏ͱΧʔϚʔΧʔ๏ ϓϩάϥϚͷͨΊͷ਺ֶษڧձ

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ઢܗܭը๏ͱΧʔϚʔΧʔ๏

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ΧʔϚʔΧʔͷ࿩͸୭ಘ

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ΧʔϚʔΧʔͷ࿩͸୭ಘ ͑ͬʂʁ

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ઢܗܭը๏ͱΧʔϚʔΧʔ๏

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ઢܗܭը๏ͱ੔਺ܭը๏

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ࣗݾ঺հ Shintaro Kaneko ‣ גࣜձࣾΤ΢ϨΧ ‣ pairsͷ։ൃશൠ୲౰ ‣ ࠷ۙ͸GoݴޠͰ։ൃ͍ͯ͠·͢ ‣ େֶ࣌୅ ‣ ࠷దԽ໰୊ͷඇઢܗܭը๏Λઐ߈ ‣ github.com/kaneshin/lbfgs ʢݹʣ ‣ ඍ෼ํఔ͕ࣜ޷͖ͩͬͨ kaneshin kaneshinth shintaro.kaneko

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ຊ೔͸౷ܭͷ࿩Ͱ͸ ͋Γ·ͤΜʂ

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ຊ೔ͷΰʔϧ

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ຊ೔ͷΰʔϧ ࠷దԽʢ਺ཧܭըʣΛ ͪΐͬͱͰ΋஌ͬͯ΋Β͏

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ຊ೔࿩͢͜ͱ

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ຊ೔࿩͢͜ͱ ΦϖϨʔγϣϯζɾϦαʔν ࠷దԽ໰୊ʢ਺ཧܭը໰୊ʣ ઢܗܭը໰୊ͱ੔਺ܭը໰୊

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ຊ೔΍Βͳ͍͜ͱ ΍Βͳ͍͜ͱ ϓϩάϥϜͰγϛϡϨʔγϣϯ ΍Γ͔ͨͬͨΜͰ͕͢ ͕࣌ؒݫ͍͠ͷͰׂѪ

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ຊ೔࿩͢͜ͱ ΦϖϨʔγϣϯζɾϦαʔν ࠷దԽ໰୊ʢ਺ཧܭը໰୊ʣ ઢܗܭը໰୊ͱ੔਺ܭը໰୊

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ΦϖϨʔγϣϯζɾϦαʔν

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ΦϖϨʔγϣϯζɾϦαʔνʢORʣ ‣ OR͸૬౰޿͍෼໺Λࢦ͢ ‣ ༷ʑͳܭըʹରͯ͠࠷΋ޮ཰తʹͳΔΑ͏ҙࢥܾఆ͢Δ ‣ ਺ֶɾ౷ܭతϞσϧɺΞϧΰϦζϜͷར༻ ‣ ʮγϛϡϨʔγϣϯʯͷཱ֬͸ORཱ͕໾ऀ ‣ ORͰ͸ݱ࣮ͷ໰୊Λ਺ཧϞσϧʹஔ͖׵͑Δ ‣ ࠷దԽɼ଴ͪߦྻɼ֬཰ͳͲ ‣ ܉ࣄతؔ৺ͱܦࡁతؔ৺͕ڧ͍෼໺

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ͭ·Γʁ

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ྫ͑͹ɼேͷ௨ۈిं

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ேͷ௨ۈిं ‣ ܦ࿏ݕࡧʹʮYahoo!࿏ઢʯ࢖͍ͬͯΔͱࢥ͍·͢ ‣ ཪଆͰ͸ʮܦ࿏ޮ཰ʯ΍ʮίετޮ཰ʯͷߟྀ ‣ Ո͔ΒӺ·Ͱʹཁ͢Δ࣌ؒΛܭࢉ ‣ ెา΍ӺͰిंΛ଴ͨͳ͍Α͏ʹग़ൃ͢Δܭը ‣ ࢁखઢͷَͷΑ͏ͳμΠϠ ‣ γϛϡϨʔγϣϯ͕Կສճ΋͞Εͨ݁Ռ

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ேͷ௨ۈిं ‣ ܦ࿏ݕࡧʹʮYahoo!࿏ઢʯ࢖͍ͬͯΔͱࢥ͍·͢ ‣ ཪଆͰ͸ʮܦ࿏ޮ཰ʯ΍ʮίετޮ཰ʯͷߟྀ ‣ Ո͔ΒӺ·Ͱʹཁ͢Δ࣌ؒΛܭࢉ ‣ ెา΍ӺͰిंΛ଴ͨͳ͍Α͏ʹग़ൃ͢Δܭը ‣ ࢁखઢͷَͷΑ͏ͳμΠϠ ‣ γϛϡϨʔγϣϯ͕Կສճ΋͞Εͨ݁Ռ ͜ΕΒ΋ORͷҰछ

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ׂͱ਎ۙʹOR

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ຊ೔࿩͢͜ͱ ΦϖϨʔγϣϯζɾϦαʔν ࠷దԽ໰୊ʢ਺ཧܭը໰୊ʣ ઢܗܭը໰୊ͱ੔਺ܭը໰୊

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࠷దԽ໰୊

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࠷దԽ໰୊͸ORͷҰཁૉ

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ઢܗܭը໰୊ ඇઢܗܭը໰୊ ੔਺ܭը໰୊ ήʔϜཧ࿦ OR σʔλ ϚΠχϯά : : : : ۚ༥޻ֶ ࠷దԽ໰୊

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࠷దԽ໰୊ͱ͸ ʢoptimization problemʣ ਺ཧܭը໰୊ͱ΋Α͹ΕΔ ʢMathematical Programming Problemʣ

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࠷దԽ໰୊ͱ͸ ༩͑ΒΕͨ৚݅ͷ΋ͱͰ ԿΒ͔ͷؔ਺Λ࠷খԽɾ࠷େԽ͢Δ໰୊

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࠷దԽ໰୊ͱ͸ ༩͑ΒΕͨ৚݅ͷ΋ͱͰ ԿΒ͔ͷؔ਺Λ࠷খԽɾ࠷େԽ͢Δ໰୊

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࠷దԽ໰୊ͱ͸

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࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰ

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࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰɼ໨తؔ਺ͷ

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࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰɼ໨తؔ਺ͷ ࠷ద஋ͱͳΔ

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࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰɼ໨తؔ਺ͷ ࠷ద஋ͱͳΔ࠷దղΛ

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࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰɼ໨తؔ਺ͷ ࠷ద஋ͱͳΔ࠷దղΛ࣮ߦՄೳྖҬ͔Β୳͢

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࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰɼ໨తؔ਺ͷ ࠷ద஋ͱͳΔ࠷దղΛ࣮ߦՄೳྖҬ͔Β୳͢

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໨తؔ਺ɾ੍໿৚݅ʹΑͬͯ ࠷దԽ໰୊͸෼ྨ͞ΕΔ

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࠷దԽ໰୊ͷ෼ྨʢ෦෼తʣ ‣ ઢܗܭը໰୊ ‣ ໨తؔ਺ɾ੍໿৚݅ͷू߹͕̍࣍ࣜͷΈ ‣ ඇઢܗܭը໰୊ ‣ ໨తؔ਺ɾ੍໿৚݅ʹඇઢܗ͕ࣜଘࡏ͢Δ ‣ ੔਺ܭը໰୊ʢࠞ߹੔਺ɾ७੔਺ʣ ‣ ੍໿৚݅ͷҰ෦·ͨ͸શͯͷม਺ͷͱΔ஋͕੔਺ ‣ ̎࣍ܭը໰୊ ‣ ໨తؔ਺͕̎࣍ࣜɼ੍໿৚݅ͷू߹͕̍࣍ࣜ etc…

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ઢܗܭը໰୊ ඇઢܗܭը໰୊ ੔਺ܭը໰୊ ήʔϜཧ࿦ OR ࠷దԽ໰୊ σʔλ ϚΠχϯά : : : : ۚ༥޻ֶ

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ઢܗܭը໰୊ ඇઢܗܭը໰୊ ੔਺ܭը໰୊ ήʔϜཧ࿦ OR ࠷దԽ໰୊ σʔλ ϚΠχϯά : : : : ۚ༥޻ֶ

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ຊ೔࿩͢͜ͱ ΦϖϨʔγϣϯζɾϦαʔν ࠷దԽ໰୊ʢ਺ཧܭը໰୊ʣ ઢܗܭը໰୊ͱ੔਺ܭը໰୊

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ઢܗܭը໰୊

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ઢܗܭը໰୊ ໨తؔ਺ɾ੍໿৚݅ͷू߹͕̍࣍ࣜͷΈ

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ྫɿੜ࢈ܭը໰୊

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̎ม਺ͷੜ࢈ܭը໰୊ ‣ M1, M2, M3ͷݪྉΛ༻͍ͯɼP1, P2Λੜ࢈͍ͯ͠Δ

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̎ม਺ͷੜ࢈ܭը໰୊ ‣ M1, M2, M3ͷݪྉΛ༻͍ͯɼP1, P2Λੜ࢈͍ͯ͠Δ ‣ ར༻ՄೳྔɿM1 ≦ 27t, M2 ≦ 45t, M3 ≦15t

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̎ม਺ͷੜ࢈ܭը໰୊ ‣ M1, M2, M3ͷݪྉΛ༻͍ͯɼP1, P2Λੜ࢈͍ͯ͠Δ ‣ ར༻ՄೳྔɿM1 ≦ 27t, M2 ≦ 45t, M3 ≦15t ‣ P1ͷੜ࢈ ‣ M1Λ2t, M2Λ8t, M3Λ3t ‣ ར५͸2ສʂ

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̎ม਺ͷੜ࢈ܭը໰୊ ‣ M1, M2, M3ͷݪྉΛ༻͍ͯɼP1, P2Λੜ࢈͍ͯ͠Δ ‣ ར༻ՄೳྔɿM1 ≦ 27t, M2 ≦ 45t, M3 ≦15t ‣ P1ͷੜ࢈ ‣ M1Λ2t, M2Λ8t, M3Λ3t ‣ ར५͸2ສʂ ‣ P2ͷੜ࢈ ‣ M1Λ6t, M2Λ6t, M3Λ1t ‣ ར५͸5ສʂ

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̎ม਺ͷੜ࢈ܭը໰୊ ‣ M1, M2, M3ͷݪྉΛ༻͍ͯɼP1, P2Λੜ࢈͍ͯ͠Δ ‣ ར༻ՄೳྔɿM1 ≦ 27t, M2 ≦ 45t, M3 ≦15t ‣ P1ͷੜ࢈ ‣ M1Λ2t, M2Λ8t, M3Λ3t ‣ ར५͸2ສʂ ‣ P2ͷੜ࢈ ‣ M1Λ6t, M2Λ6t, M3Λ1t ‣ ར५͸5ສʂ ར५Λ࠷େԽ͍ͨ͠

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̎ม਺ͷੜ࢈ܭը໰୊ ݪྉʘ੡඼ P1 P2 ར༻Մೳྔ M1 2 6 27 M2 8 6 45 M3 3 1 15 ར५ 2 5

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̎ม਺ͷੜ࢈ܭը໰୊ ݪྉʘ੡඼ P1 P2 ར༻Մೳྔ M1 2 6 27 M2 8 6 45 M3 3 1 15 ར५ 2 5

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̎ม਺ͷੜ࢈ܭը໰୊ ੡඼P1, P2ͷੜ࢈ྔΛมྔ (x1, x2) ͱ͢Δͱ Λ࠷େԽͤ͞Δ࠷దԽ໰୊ͱͳΔ

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̎ม਺ͷੜ࢈ܭը໰୊ ݪྉʘ੡඼ P1 P2 ར༻Մೳྔ M1 2 6 27 M2 8 6 45 M3 3 1 15 ར५ 2 5

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̎ม਺ͷੜ࢈ܭը໰୊ ੍໿৚݅͸ར༻Մೳͳݪྉͷྔͱ੡඼͸ඇෛ

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̎ม਺ͷੜ࢈ܭը໰୊ ੍໿৚݅͸ར༻Մೳͳݪྉͷྔͱ੡඼͸ඇෛ

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̎ม਺ͷੜ࢈ܭը໰୊ ઢܗܭը໰୊ͱͯ͠ఆࣜԽ

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̎ม਺ͷੜ࢈ܭը໰୊ ઢܗܭը໰୊ͱͯ͠ఆࣜԽ ͜ͷ໰୊ͷ࠷దղΛٻΊ͍ͨ

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ͱ͜ΖͰ

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੡඼͸࿈ଓ஋͔ʁ

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੔਺ܭը໰୊

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੔਺ܭը໰୊ ੍໿৚݅ͷҰ෦·ͨ͸શͯͷม਺ͷͱΔ஋͕੔਺

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ྫɿੜ࢈ܭը໰୊ʢ੔਺ʣ

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̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ ‣ M1, M2, M3ͷݪྉΛ༻͍ͯɼP1, P2Λੜ࢈͍ͯ͠Δ ‣ ར༻ՄೳྔɿM1 ≦ 27t, M2 ≦ 45t, M3 ≦15t ‣ P1ͷੜ࢈ ‣ M1Λ2t, M2Λ8t, M3Λ3t ‣ ར५͸2ສʂ ‣ P2ͷੜ࢈ ‣ M1Λ6t, M2Λ6t, M3Λ1t ‣ ར५͸5ສʂ ར५Λ࠷େԽ͍ͨ͠

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̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ ‣ M1, M2, M3ͷݪྉΛ༻͍ͯɼP1, P2Λੜ࢈͍ͯ͠Δ ‣ ར༻ՄೳྔɿM1 ≦ 27t, M2 ≦ 45t, M3 ≦15t ‣ P1ͷੜ࢈ɹݸ਺ͰΫϨ ‣ M1Λ2t, M2Λ8t, M3Λ3t ‣ ར५͸2ສʂ ‣ P2ͷੜ࢈ɹݸ਺ͰΫϨ ‣ M1Λ6t, M2Λ6t, M3Λ1t ‣ ར५͸5ສʂ ར५Λ࠷େԽ͍ͨ͠

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̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ ઢܗܭը໰୊ͱͯ͠ఆࣜԽ

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̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ ઢܗܭը໰୊ͱͯ͠ఆࣜԽ ੔਺ͰΫϨ ੔਺ͰΫϨ

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̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ ੔਺ܭը໰୊ͱͯ͠ఆࣜԽ

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̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ ੔਺ܭը໰୊ͱͯ͠ఆࣜԽ ͜ͷ໰୊ͷ࠷దղΛٻΊ͍ͨ

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άϥϑͰΈΔղ๏ ԼهΛάϥϑͰදݱ͢Δ

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ઢܗܭը໰୊ͷ৔߹

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ઢܗܭը໰୊ͷ৔߹

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ઢܗܭը໰୊ͷ৔߹

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ઢܗܭը໰୊ͷ৔߹ ࠷దղ

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੔਺ܭը໰୊ͷ৔߹

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੔਺ܭը໰୊ͷ৔߹ ࠷దղ?

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੔਺ܭը໰୊ͷ৔߹ ࠷దղ!

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ઢܗܭըͱ੔਺ܭըͷҧ͍ ‣ ઢܗܭը໰୊ͷ࠷దղ ‣ ୯ମ๏ͱ͍͏ɺลΛḷ͍ͬͯ͘ํ๏͕࣮༻త ‣ ଟ߲ࣜ࣌ؒͰղ͚ͳ͍ͱ͖͕͋Δ ‣ ಺఺๏ͱ͍͏ղ๏΋ଘࡏ͢Δ ‣ ࣮ߦྖҬͷ಺෦͔ΒղΛ୳ࡧ͢Δ ‣ ੔਺ܭըͷ࠷దղ ‣ ઢܗܭըͰಘΒΕͨ࠷దղʹ͍ۙ੔਺ͷ఺͕࠷దղʹͳΒͳ͍ ৔߹͕͋Δ ಺఺๏͸ ϫγ͕ҭͯͨ

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·ͱΊ

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·ͱΊ ࠷దԽ໰୊͸ ͳΜͱͳ͘ཧղͯ͠΋Β͑ͨͰ͠ΐ͏͔ʁ

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·ͱΊ - ࠷ޙʹ ‣ ͜ͷ෼໺͸·ͩ·ͩϓϩάϥϜͷࢿ࢈͕গͳ͍ ‣ ਺ཧϞσϧʹམͱͨ͢Ίʹ਺ֶͷ஌͕ࣝͳ͍ͱ೉͍͠ ‣ ઐ໳తͳاۀ͕ಠ઎͍ͯ͠Δײ͕͋Δ ‣ ͦΕͧΕͷܭը໰୊͸ߟ͑Δ͜ͱ͕ҧ͏ ‣ ݚڀ͍ͨ͠৔߹͸෼໺ΛߜΔ΂͖

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·ͱΊ ͜ΕҎ্ͷ࠷దԽ໰୊ͷ࿩΍ γϛϡϨʔγϣϯͳͲ͸ ୭͔͕΍ͬͯ͘ΕΔͱ৴͍ͯ͡·͢ʂ