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ػցֶशºηΩϡϦςΟ Ø %PT߈ܸΛػցֶशͰݕ஌ Ø Ϛϧ΢ΣΞΛػցֶशͰݕ஌ ػցֶशΛηΩϡϦςΟʹԠ༻ ػցֶशϞσϧࣗମͷηΩϡϦςΟ ػցֶशͰߏஙͨ͠ϞσϧʢػցֶशϞσϧʣΛ૊ΈࠐΜͩγεςϜͰ͸ ैདྷͷγεςϜͱ͸ҟͳͬͨݻ༗ͷ߈ܸ͕ͳ͞ΕΔϦεΫ͕༗Δ ͦͷͨΊɼػցֶशϞσϧಛ༗ͷηΩϡϦςΟରࡦ͕ඞཁ 2

"EWFSTBSJBM"UUBDL ϞσϧΛὃ͢߈ܸ .PEFM&YUSBDUJPO ϞσϧΛ౪Ή߈ܸ .PEFM*OWFSTJPO ֶशσʔλʹؔ͢Δ৘ใΛ෮ݩ͢Δ߈ܸ

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"EWFSTBSJBM"UUBDL ܇࿅͞ΕͨػցֶशϞσϧΛὃ͢߈ܸ ݩͷը૾ ఢରతઁಈ "EWFSTBSJBM&YBNQMF ʢςφΨβϧʣ (PPEGFMMPX FUBM ΑΓҾ༻ 5

ఢରతઁಈΛݟ͚ͭΔͨΊʹ͸ argmin " , s.t. + ≠ argmin " , s.t. + = # ಛఆͷΫϥεʹޡ෼ྨͤ͞Δ͜ͱΛ໨తͱͨ͠ "EWFSTBSJBM"UUBDL ޡ෼ྨͤ͞ΔΫϥε͸ࢦఆͤͣ ͱʹ͔͘Ͳ͔͜ͷΫϥεʹޡ෼ྨͤ͞Δ͜ͱΛ ໨తͱͨ͠"EWFSTBSJBM"UUBDL 6

୅දతͳ"EWFSTBSJBM"UUBDL (PPEGFMMPX FUBM !"# = + sign ∇ ℓ , /PUBUJPO ɿઁಈڧ౓ʢͲΕ͙Β͍ઁಈΛڧ͘༩͑Δ͔Λද͢ʣ ℓɿଛࣦؔ਺ 3FNBSL '(4.͸Ұ౓ͷޯ഑্ঢͰ࠷దԽΛऴྃ͢Δ͕ɼ͜ΕΛෳ਺ճʹ֦ுͨ͠ͷ͕#*.Ͱɼ CBMMʹऩ·ΔΑ͏ͳ੍໿ͷ΋ͱͰޯ഑্ঢΛ܁Γฦ͢ɽ '(4.΋#*.΋! ϊϧϜͰͷઁಈΛੜ੒͓ͯ͠Γɼ%FFQGPPM͸͜ͷϊϧϜΛ" ʹɼ $8͸# ʹͨ͠΋ͷͱݟΔ͜ͱ͕Ͱ͖Δɽ 7

8IJUF#PY4FUUJOHʹ͓͚Δఢରతઁಈͷݟ͚ͭํ '(4.BUUBDL (PPEGFMMPX FUBM #*. ,VSBLJO FUBM BUUBDL.*'(4. %POHFUBM %FFQGPPM .PPTBWJ%F[GPPMJ FUBM /FXUPO'PPM +BOHFUBM +4." 1BQFSOPU FUBM (SBEJFOU #BTFE 0CKFDUJWF 'VODUJPO #BTFE $8 $BSMJOJ FUBM &"% $IFOFUBM 0QU.BSHJO 8BSSFOFUBM ˞8IJUF#PY4FUUJOHɿଛࣦؔ਺ͷޯ഑͕ܭࢉͰ͖ͨΓɼϞσϧͷDPOGJEFODFTDPSFΛ஌Δ͜ͱ͕Ͱ͖Δઃఆ 8

"EWFSTBSJBM"UUBDL΁ͷରࡦɿ"EWFSTBSJBM5SBJOJOH Ϟσϧͷֶश࣌ʹBEWFSTBSJBMFYBNQMFTͰl༧๷઀छz͓͚ͯ͠͹ BEWFSTBSJBMFYBNQMFTʹର͢Δϩόετੑ΋্͕ΔΜ͡Όͳ͍͔ʁ 9 ℓ!"# , () ≔ ℓ !"#, !"# + 1 − ℓ , *EFB 3FTVMU n ࣮ࡍʹBEWFSTBSJBMFYBNQMFTʹର͢Δϩόετੑ͸޲্͢Δ ҰํͰʜ n ֶशʹཁ͢Δ͕࣌ؒ૿͑Δ n "EWFSTBSJBMFYBNQMFT΁ͷϩόετੑ͸BEWFSTBSJBMFYBNQMFTͷ࡞Γํʹґଘ͢Δ n ѱҙͷͳ͍ϊΠζ͕৐ͬͨը૾Λ͏·͘෼ྨͰ͖ͳ͘ͳΔ ͱ͍ͬͨ໰୊΋ൃੜ͢Δ

"EWFSTBSJBM"UUBDL ϞσϧΛὃ͢߈ܸ .PEFM&YUSBDUJPO ϞσϧΛ౪Ή߈ܸ .PEFM*OWFSTJPO ֶशσʔλʹؔ͢Δ৘ใΛ෮ݩ͢Δ߈ܸ

.PEFM&YUSBDUJPOɿϞσϧΛ౪Ή & "SDIJUFDUVSF %FDJTJPO#PVOEBSZ 'VODUJPOBMJUZ ϨΠϠʔͷ਺΍χϡʔϩϯͷ਺ ͞Βʹ׆ੑԽؔ਺ͳͲͷ Ϟσϧͷߏ଄Λ౪ΉλΠϓ ϞσϧͷܾఆڥքΛ ౪ΉλΠϓ ϞσϧʹΑΔ ೖྗͱग़ྗͷରԠؔ܎Λ ౪ΉλΠϓ 11

.PEFM&YUSBDUJPOͷओཁͳΞϓϩʔν $ , $ $%& ' : ℝ( → ֶश ), ) )%& * +: ℝ( → ֶश 0SJHJOBM.PEFM 4VCTUJUVUF.PEFM "1*ʹΑͬͯฦ͞ΕΔ Λ໨తม਺ͱͯ͠ར༻͢Δ͜ͱͰ୅ཧϞσϧΛߏங͢Δɽ ୅දతͳݚڀͱͯ͠$PSSFJB4JMWBFUBM ΍0SFLPOEZ FUBM ͕ڍ͛ΒΕΔɽ 12

4VCTUJUVUF.PEFMʹؔ͢Δ߈๷ %FGFODF 0GGFODF "1*ୟ͚Δճ਺Λ੍ݶ গͳ͍Ԡ౴ճ਺Ͱ΋ 4VCTUJUVUF.PEFMΛߏங͢Δํ๏͕ఏҊ͞ΕΔ 0SFLPOEZ FUBM 3PTFOCFSHFUBM 13

"EWFSTBSJBM"UUBDL ϞσϧΛὃ͢߈ܸ .PEFM&YUSBDUJPO ϞσϧΛ౪Ή߈ܸ .PEFM*OWFSTJPO ֶशσʔλʹؔ͢Δ৘ใΛ෮ݩ͢Δ߈ܸ

ʢ޿ٛͷʣ.PEFM*OWFSTJPO"UUBDL ֶशσʔλʹؔ͢Δ৘ใΛ෮ݩ͢Δ߈ܸ 1SPQFSUZ*OGFSFODF"UUBDL .PEFM*OWFSTJPO"UUBDL ద౰ͳ ͕Ϟσϧͷֶशσʔληοτʹؚ·Ε͍ͯΔ͔ʁ Λ໌Β͔ʹ͢Δ߈ܸ ֶशσʔληοτʹؔ͢Δੑ࣭Λਪଌ͢Δ߈ܸ FH൅͕௕͍ਓ΍υϨεΛண͍ͯΔਓ͕ ੑผ෼ྨͷϞσϧͷֶशσʔλʹؚ·Ε͍ͯΔ͔ʁ ֶशσʔλΛ෮ݩ͢Δ߈ܸ ˞.PEFM*OWFSTJPO"UUBDLʹ.FNCFSTIJQ*OGFSFODF΍1SPQFSUZ*OGFSFODFΛؚΊΔ͔ʹ͸ॾઆ͋Γ 15

("/Λ༻͍ͨ.PEFM*OWFSTJPO ,VTBOP FUBM ิॿσʔληοτ ) , )%& * , , ∼$$( ℱ′ ֶशσʔληοτ $ , $ $%& ' , ∼$$( ℱ ΫΤϦ & ,, … * , Ԡ౴ (& ,), … * , (FOFSBUPS Ͱ ℱ ʢͬΆ͍΋ͷʣΛۙࣅ αϯϓϦϯά (FOFSBUPS͔Β ֶशσʔλͬΆ͍΋ͷΛੜ੒ 16

σʔλͷҰ෦͔Β࢒Γͷ෦෼Λ෮ݩ ֶशσʔλ ∈ ℝ( ͔ΒҰ෦ͷಛ௃ྔ͚ͩΛൈ͖ग़ͯ͠ ࡞੒ͨ͠ϕΫτϧ " ∈ ℝ-, < ͔Β ࢒Γͷಛ௃ྔΛ෮ݩ͢Δ ࿦จͰ͸ " ͸࿙Ӯͯ͠΋໰୊ͳ͍ηϯγςΟϒͰͳ͍ಛ௃ྔͰ ࢒Γͷಛ௃ྔ͸ηϯγςΟϒͳಛ௃ྔͩͱ૝ఆ͍ͯ͠Δ ("/Λ༻͍ͨ.PEFM*OWFSTJPO ;IBOFUBM 17

͍΍ɼϜζ͘Ͷʜʁ ./*45Λֶशσʔλͱͨ͠෼ྨϞσϧʹରͯ͠.PEFM *OWFSTJPO"UUBDLΛߦͬͨ݁Ռ ิॿσʔλ͸खॻ͖จࣈͷ਺ࣈͱΞϧϑΝϕοτ ࣮ࡍʹ෮ݩͯ͠Έͨ݁Ռ 18

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ػցֶशϞσϧ΁ͷ߈ܸʹؔ͢Δݚڀ ΍ͬͺΓ"EWFSTBSJBM"UUBDL͕μϯτπͳײ͡ ݚڀ͞ΕͯΔײ ֶशσʔλΛ͢΂ͯ෮ݩ͢ΔλΠϓͷ.PEFM*OWFSTJPO"UUBDL͕Ұ൪೉ͦ͠͏ ʢ࣮ࡍʹ೉͔ͬͨ͠ʣ ߈ܸͷ೉͠͞ n "EWFSTBSJBM"UUBDL͸σʔλ͕ߴ࣍ݩʹͳΔͱෆՄආతʹੜ͡ΔͨΊ ຊ࣭తʹରࡦ͕ࠔ೉ʁ n .PEFM&YUSBDUJPO"UUBDL͸ΫΤϦ੍ݶ͕༗ޮʹࢥ͑Δ͕ ΫΤϦ੍ݶͷ΋ͱͰ΋͋Δఔ౓ͷϞσϧෳ੡͕ग़དྷ͓ͯΓ ࠓޙͷಈ޲ʹ஫໨ ରࡦͷ೉͠͞ 20

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