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〜

Transcript

1. ઢܗܭը๏ͱΧʔϚʔΧʔ๏ ϓϩάϥϚͷͨΊͷ਺ֶษڧձ

2. ઢܗܭը๏ͱΧʔϚʔΧʔ๏

3. ΧʔϚʔΧʔͷ࿩͸୭ಘ

4. ΧʔϚʔΧʔͷ࿩͸୭ಘ ͑ͬʂʁ

5. ઢܗܭը๏ͱΧʔϚʔΧʔ๏

6. ઢܗܭը๏ͱ੔਺ܭը๏

7. ࣗݾ঺հ Shintaro Kaneko ‣ גࣜձࣾΤ΢ϨΧ ‣ pairsͷ։ൃશൠ୲౰ ‣ ࠷ۙ͸GoݴޠͰ։ൃ͍ͯ͠·͢ ‣

   େֶ࣌୅ ‣ ࠷దԽ໰୊ͷඇઢܗܭը๏Λઐ߈ ‣ github.com/kaneshin/lbfgs ʢݹʣ ‣ ඍ෼ํఔ͕ࣜ޷͖ͩͬͨ kaneshin kaneshinth shintaro.kaneko

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

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

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

11. Copyright © 2009-2015 eureka, Inc. All rights reserved. ‣ pairs͸ϚονϯάΞϧΰϦζϜ͕ඇৗʹॏཁ

    ‣ ๛෋ͳϢʔβʔͷσʔλͷ׆༻ ‣ ΤϯδχΞ͕Ϗδωε໘͔ΒΞϧΰϦζϜΛߟ͑Δ ‣ ࣾ಺ษڧձͰRษڧʴ౷ܭֶͷษڧձ΋։࠵ ‣ ϏοάσʔλΛਖ਼͍͠஌ࣝͰ࢖༻͢Δ ‣ ౷ܭΛؒҧͬͨೝࣝͰ࢖༻͠ͳ͍ʢภͬͨσʔλΛ࢖Θͳ͍

12. ຊ೔͸౷ܭͷ࿩Ͱ͸ ͋Γ·ͤΜʂ

13. ຊ೔ͷΰʔϧ

14. ຊ೔ͷΰʔϧ ࠷దԽʢ਺ཧܭըʣΛ ͪΐͬͱͰ΋஌ͬͯ΋Β͏

15. ຊ೔࿩͢͜ͱ

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

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

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

19. ΦϖϨʔγϣϯζɾϦαʔν

20. ΦϖϨʔγϣϯζɾϦαʔνʢORʣ ‣ OR͸૬౰޿͍෼໺Λࢦ͢ ‣ ༷ʑͳܭըʹରͯ͠࠷΋ޮ཰తʹͳΔΑ͏ҙࢥܾఆ͢Δ ‣ ਺ֶɾ౷ܭతϞσϧɺΞϧΰϦζϜͷར༻ ‣ ʮγϛϡϨʔγϣϯʯͷཱ֬͸ORཱ͕໾ऀ ‣

    ORͰ͸ݱ࣮ͷ໰୊Λ਺ཧϞσϧʹஔ͖׵͑Δ ‣ ࠷దԽɼ଴ͪߦྻɼ֬཰ͳͲ ‣ ܉ࣄతؔ৺ͱܦࡁతؔ৺͕ڧ͍෼໺

21. ͭ·Γʁ

22. ྫ͑͹ɼேͷ௨ۈిं

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

    ࢁखઢͷَͷΑ͏ͳμΠϠ ‣ γϛϡϨʔγϣϯ͕Կສճ΋͞Εͨ݁Ռ

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

    ࢁखઢͷَͷΑ͏ͳμΠϠ ‣ γϛϡϨʔγϣϯ͕Կສճ΋͞Εͨ݁Ռ ͜ΕΒ΋ORͷҰछ

25. ׂͱ਎ۙʹOR

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

27. ࠷దԽ໰୊

28. ࠷దԽ໰୊͸ORͷҰཁૉ

29. ઢܗܭը໰୊ ඇઢܗܭը໰୊ ੔਺ܭը໰୊ ήʔϜཧ࿦ OR σʔλ ϚΠχϯά : : :

    : ۚ༥޻ֶ ࠷దԽ໰୊

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

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

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

33. ࠷దԽ໰୊ͱ͸

34. ࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰ

35. ࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰɼ໨తؔ਺ͷ

36. ࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰɼ໨తؔ਺ͷ ࠷ద஋ͱͳΔ

37. ࠷దԽ໰୊ͱ͸ ੍໿৚݅ͷ΋ͱͰɼ໨తؔ਺ͷ ࠷ద஋ͱͳΔ࠷దղΛ

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

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

40. ໨తؔ਺ɾ੍໿৚݅ʹΑͬͯ ࠷దԽ໰୊͸෼ྨ͞ΕΔ

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…

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

    : : ۚ༥޻ֶ

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

    : : ۚ༥޻ֶ

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

45. ઢܗܭը໰୊

46. ઢܗܭը໰୊ ໨తؔ਺ɾ੍໿৚݅ͷू߹͕̍࣍ࣜͷΈ

47. ྫɿੜ࢈ܭը໰୊

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

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

    M2 ≦ 45t, M3 ≦15t

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

    M2 ≦ 45t, M3 ≦15t ‣ P1ͷੜ࢈ ‣ M1Λ2t, M2Λ8t, M3Λ3t ‣ ར५͸2ສʂ

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

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

53. ̎ม਺ͷੜ࢈ܭը໰୊ ݪྉʘ੡඼ P1 P2 ར༻Մೳྔ M1 2 6 27 M2

    8 6 45 M3 3 1 15 ར५ 2 5

54. ̎ม਺ͷੜ࢈ܭը໰୊ ݪྉʘ੡඼ P1 P2 ར༻Մೳྔ M1 2 6 27 M2

    8 6 45 M3 3 1 15 ར५ 2 5

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

56. ̎ม਺ͷੜ࢈ܭը໰୊ ݪྉʘ੡඼ P1 P2 ར༻Մೳྔ M1 2 6 27 M2

    8 6 45 M3 3 1 15 ར५ 2 5

57. ̎ม਺ͷੜ࢈ܭը໰୊ ੍໿৚݅͸ར༻Մೳͳݪྉͷྔͱ੡඼͸ඇෛ

58. ̎ม਺ͷੜ࢈ܭը໰୊ ੍໿৚݅͸ར༻Մೳͳݪྉͷྔͱ੡඼͸ඇෛ

59. ̎ม਺ͷੜ࢈ܭը໰୊ ઢܗܭը໰୊ͱͯ͠ఆࣜԽ

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

61. ͱ͜ΖͰ

62. ੡඼͸࿈ଓ஋͔ʁ

63. ੔਺ܭը໰୊

64. ੔਺ܭը໰୊ ੍໿৚݅ͷҰ෦·ͨ͸શͯͷม਺ͷͱΔ஋͕੔਺

65. ྫɿੜ࢈ܭը໰୊ʢ੔਺ʣ

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

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

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 ੔਺ ੔਺

69. ̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ ઢܗܭը໰୊ͱͯ͠ఆࣜԽ

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

71. ̎ม਺ͷੜ࢈ܭը໰୊ʢ੔਺ʣ ੔਺ܭը໰୊ͱͯ͠ఆࣜԽ

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

73. άϥϑͰΈΔղ๏ ԼهΛάϥϑͰදݱ͢Δ

74. None
75. None

76. ઢܗܭը໰୊ͷ৔߹

77. ઢܗܭը໰୊ͷ৔߹

78. ઢܗܭը໰୊ͷ৔߹

79. ઢܗܭը໰୊ͷ৔߹ ࠷దղ

80. ੔਺ܭը໰୊ͷ৔߹

81. ੔਺ܭը໰୊ͷ৔߹ ࠷దղ?

82. ੔਺ܭը໰୊ͷ৔߹ ࠷దղ!

83. ઢܗܭըͱ੔਺ܭըͷҧ͍ ‣ ઢܗܭը໰୊ͷ࠷దղ ‣ ୯ମ๏ͱ͍͏ɺลΛḷ͍ͬͯ͘ํ๏͕࣮༻త ‣ ଟ߲ࣜ࣌ؒͰղ͚ͳ͍ͱ͖͕͋Δ ‣ ಺఺๏ͱ͍͏ղ๏΋ଘࡏ͢Δ ‣

    ࣮ߦྖҬͷ಺෦͔ΒղΛ୳ࡧ͢Δ ‣ ੔਺ܭըͷ࠷దղ ‣ ઢܗܭըͰಘΒΕͨ࠷దղʹ͍ۙ੔਺ͷ఺͕࠷దղʹͳΒͳ͍ ৔߹͕͋Δ ಺఺๏͸ ϫγ͕ҭͯͨ

84. ·ͱΊ

85. ·ͱΊ ࠷దԽ໰୊͸ ͳΜͱͳ͘ཧղͯ͠΋Β͑ͨͰ͠ΐ͏͔ʁ

86. ·ͱΊ - ࠷ޙʹ ‣ ͜ͷ෼໺͸·ͩ·ͩϓϩάϥϜͷࢿ࢈͕গͳ͍ ‣ ਺ཧϞσϧʹམͱͨ͢Ίʹ਺ֶͷ஌͕ࣝͳ͍ͱ೉͍͠ ‣ ઐ໳తͳاۀ͕ಠ઎͍ͯ͠Δײ͕͋Δ ‣

    ͦΕͧΕͷܭը໰୊͸ߟ͑Δ͜ͱ͕ҧ͏ ‣ ݚڀ͍ͨ͠৔߹͸෼໺ΛߜΔ΂͖

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

88. Thank you :)

89. ͓·͚ - ಺఺๏ʢΧʔϚʔΧʔ๏ʣ