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
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
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