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〜

カーマーカー法はやめて「線形計画法と整数計画法」です。

Transcript

1. ઢܗܭը๏ͱΧʔϚʔΧʔ๏
   ϓϩάϥϚͷͨΊͷ਺ֶษڧձ
   

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2. ઢܗܭը๏ͱΧʔϚʔΧʔ๏
   

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3. ΧʔϚʔΧʔͷ࿩͸୭ಘ
   

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4. ΧʔϚʔΧʔͷ࿩͸୭ಘ
   ͑ͬʂʁ
   

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5. ઢܗܭը๏ͱΧʔϚʔΧʔ๏
   

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6. ઢܗܭը๏ͱ੔਺ܭը๏
   

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7. ࣗ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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8. ࣗݾ঺հ
   Shintaro Kaneko
   ‣ גࣜձࣾΤ΢ϨΧ
   ‣ pairsͷ։ൃશൠ୲౰
   ‣ ࠷ۙ͸GoݴޠͰ։ൃ͍ͯ͠·͢
   ‣ େֶ࣌୅
   ‣ ࠷దԽ໰୊ͷඇઢܗܭը๏Λઐ߈
   ‣ github.com/kaneshin/lbfgs ʢݹʣ
   ‣ ඍ෼ํఔ͕ࣜ޷͖ͩͬͨ
   kaneshin
   kaneshinth
   shintaro.kaneko
   ٱ͠ৼΓͷ਺ֶʂ
   

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9. Copyright © 2009-2015 eureka, Inc. All rights reserved.
   

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10. Copyright © 2009-2015 eureka, Inc. All rights reserved.
    

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11. Copyright © 2009-2015 eureka, Inc. All rights reserved.
    ‣ pairs͸ϚονϯάΞϧΰϦζϜ͕ඇৗʹॏཁ
    ‣ ๛෋ͳϢʔβʔͷσʔλͷ׆༻
    ‣ ΤϯδχΞ͕Ϗδωε໘͔ΒΞϧΰϦζϜΛߟ͑Δ
    ‣ ࣾ಺ษڧձͰRษڧʴ౷ܭֶͷษڧձ΋։࠵
    ‣ ϏοάσʔλΛਖ਼͍͠஌ࣝͰ࢖༻͢Δ
    ‣ ౷ܭΛؒҧͬͨೝࣝͰ࢖༻͠ͳ͍ʢภͬͨσʔλΛ࢖Θͳ͍
    

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12. ຊ೔͸౷ܭͷ࿩Ͱ͸
    ͋Γ·ͤΜʂ
    

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13. ຊ೔ͷΰʔϧ
    

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14. ຊ೔ͷΰʔϧ
    ࠷దԽʢ਺ཧܭըʣΛ
    ͪΐͬͱͰ΋஌ͬͯ΋Β͏
    

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15. ຊ೔࿩͢͜ͱ
    

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

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17. ຊ೔΍Βͳ͍͜ͱ
    ΍Βͳ͍͜ͱ
    ϓϩάϥϜͰγϛϡϨʔγϣϯ
    ΍Γ͔ͨͬͨΜͰ͕͢
    ͕࣌ؒݫ͍͠ͷͰׂѪ
    

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

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19. ΦϖϨʔγϣϯζɾϦαʔν
    

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20. 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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21. ͭ·Γʁ
    

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22. ྫ͑͹ɼேͷ௨ۈిं
    

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

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

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25. ׂͱ਎ۙʹOR
    

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

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27. ࠷దԽ໰୊
    

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28. ࠷దԽ໰୊͸ORͷҰཁૉ
    

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

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

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

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

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33. ࠷దԽ໰୊ͱ͸
    

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34. ࠷దԽ໰୊ͱ͸
    ੍໿৚݅ͷ΋ͱͰ
    

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35. ࠷దԽ໰୊ͱ͸
    ੍໿৚݅ͷ΋ͱͰɼ໨తؔ਺ͷ
    

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

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

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

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

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40. ໨తؔ਺ɾ੍໿৚݅ʹΑͬͯ
    ࠷దԽ໰୊͸෼ྨ͞ΕΔ
    

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

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

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

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45. ઢܗܭը໰୊
    

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46. ઢܗܭը໰୊
    ໨తؔ਺ɾ੍໿৚݅ͷू߹͕̍࣍ࣜͷΈ
    

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47. ྫɿੜ࢈ܭը໰୊
    

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

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

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50. ̎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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51. ̎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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52. ̎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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53. ̎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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54. ̎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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55. ̎ม਺ͷੜ࢈ܭը໰୊
    ੡඼P1, P2ͷੜ࢈ྔΛมྔ (x1, x2) ͱ͢Δͱ
    Λ࠷େԽͤ͞Δ࠷దԽ໰୊ͱͳΔ
    

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56. ̎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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57. ̎ม਺ͷੜ࢈ܭը໰୊
    ੍໿৚݅͸ར༻Մೳͳݪྉͷྔͱ੡඼͸ඇෛ
    

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58. ̎ม਺ͷੜ࢈ܭը໰୊
    ੍໿৚݅͸ར༻Մೳͳݪྉͷྔͱ੡඼͸ඇෛ
    

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59. ̎ม਺ͷੜ࢈ܭը໰୊
    ઢܗܭը໰୊ͱͯ͠ఆࣜԽ
    

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60. ̎ม਺ͷੜ࢈ܭը໰୊
    ઢܗܭը໰୊ͱͯ͠ఆࣜԽ
    ͜ͷ໰୊ͷ࠷దղΛٻΊ͍ͨ
    

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61. ͱ͜ΖͰ
    

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62. ੡඼͸࿈ଓ஋͔ʁ
    

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63. ੔਺ܭը໰୊
    

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64. ੔਺ܭը໰୊
    ੍໿৚݅ͷҰ෦·ͨ͸શͯͷม਺ͷͱΔ஋͕੔਺
    

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65. ྫɿੜ࢈ܭը໰୊ʢ੔਺ʣ
    

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66. ̎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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67. ̎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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68. ̎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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69. ̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ
    ઢܗܭը໰୊ͱͯ͠ఆࣜԽ
    

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

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71. ̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ
    ੔਺ܭը໰୊ͱͯ͠ఆࣜԽ
    

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

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73. άϥϑͰΈΔղ๏
    ԼهΛάϥϑͰදݱ͢Δ
    

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76. ઢܗܭը໰୊ͷ৔߹
    

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77. ઢܗܭը໰୊ͷ৔߹
    

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78. ઢܗܭը໰୊ͷ৔߹
    

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79. ઢܗܭը໰୊ͷ৔߹
    ࠷దղ
    

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80. ੔਺ܭը໰୊ͷ৔߹
    

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81. ੔਺ܭը໰୊ͷ৔߹
    ࠷దղ?
    

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82. ੔਺ܭը໰୊ͷ৔߹
    ࠷దղ!
    

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83. 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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84. ·ͱΊ
    

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85. ·ͱΊ
    ࠷దԽ໰୊͸
    ͳΜͱͳ͘ཧղͯ͠΋Β͑ͨͰ͠ΐ͏͔ʁ
    

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

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

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88. Thank you :)
    

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89. ͓·͚ - ಺఺๏ʢΧʔϚʔΧʔ๏ʣ
    

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