最优化与KKT条件

1ABriefLectureNotesonOptimizationforMicroeconomicAnalysis120031Dixit,Chiang,[e-mail]2A.…………………………………………………..………..3B.…………………………………………………………………3B.1.…………………………………………………..3B.1.1…………………………………………………4B.2.……………………………………………………..6B.2.1…………………………………………………8B.2.2…………………………………………………8B.2.3………………………………………………………9C.……………………………………..10D.…………………………..11E.………………………………………………………………..14E.1…………………………………………………14E.1.1…………………………………………………..14E.1.2…………………………………………………….16E.2……………………………………………….17E.3………………………………………………….21F.……………………………………………………………………22F.1.1…………………………………………………………22F.1.2………………………………………………………22F.1.3.………………………………………………………23F.2.……………………………………………………………..24F.3.……………………………………………..25G.…………………………………………………………28G.1.…………………………………………………………..28G.1.1…………………………………………………………28G.1.2.…………………………………………………28G.1.3……………………………………………………28G.2.……………………………………………..29G.2.1.…………………………………………………29G.2.2…………………………………………………30G.3……………………………………………………………….31H.……………………………………………………………33H.1.…………………………………………………………33H.2.……………………………………...……………………34I.…………………………………………….…………………….36I.1.…………………………………………………………36I.2.……………………………………………………………37I.3.………………………………………………39I.4.……………………………………………………403A.cxgtsxfx=)(..)(max)(xfx)(xgS{x|g(x)=c}Sxf(x)Sf(x)=kSx2Sgradff(x)=kx1A.1B.gradfB.1.dxxfxdf⋅=)()('xx0)('xf0dx0)(xdfx0)('xf0dx0)(xdf4f(x)0)('=xfdxfxdfx⋅=)(xf)(xfx),,1nffLdxx)(xfdxnnnnxdxfdxfdxdxffdxfxdf++==⋅=LML1111),,()(B.1.1x,ynnnnyxyxyyxxyx++==⋅LML1111),,(αcosyxyx=⋅yxyx⋅=αcosαx,yyx,x,yxy20πα≤0⋅yx2πα=0=⋅yxx,yπαπ≤20⋅yxxfdxdxx),(21dxdxdx=xdxx5x2dxdx2xdx1x1B.1x,2211Ixpxp=+0,021≥≥xxx2fxxdxx1B.2Ipp,,2102211=+dxpdxp21,dxdx1dx2dx2dx1dxdxxfkxf=)(xxf(x)xf6xxfdxfxdfx⋅=)(xfdx2π0)(xdfx)(xfxfdx2π0)(xdfx)(xfxfdx0)(=xdfxxx0=⋅dxfxdxxfxdxxf0)(xdfx0)(=xfxB.2.xdxdxxfdxx1x3xfdxx2xfdx0)(=xdfdxxfx2x27x2x1x2x3x1B.3cxG=)(xdx011=++nndxGdxGLc0),,(11=⋅nndxdxGGML0=⋅dxGx),,(1nxGGGL=xdxdxxGx0=⋅dxfxxfdxxxfxGλxxGfλ=λ=xxGfxfxG)(xGxcxG=)(x0=⋅dxGxdxxGdxcxG=)(x0=⋅dxfxxfdxdxkxf=)(xkxf=)(cxG=)(xfxG8xxGfλ=gradf)('xfnxfB.2.1m)(xGcmnmncxxGcxxG==),,(),,(1111LML∂∂∂∂∂∂∂∂≡∂∂≡nmmnnmxxGxGxGxGxxGGG,,,,),,(),,(111111LMLLL)(xGB.2.2i)0),(=yxfyx,),(00yx0≠yfyx)(xyφ=yxffdxdy−=ii)xn),(00yx0≠yfynx)(xyφ=niffxyyii,,1,L=−=∂∂0),(=yxf011=+++dyfdxfdxfynnLiii)ymy),(yxfm0),,;,,(0),,;,,(11111==mnmmnyyxxfyyxxfLLMLL9∂∂∂∂∂∂∂∂=∂∂=mmmmmmyyfyfyfyfyyfff,,,,),,(),,(111111LMLLLmff,,1Lmyy,,1Lm×m),(00yxyf0det≠yfyx)(xyφ=)(xφ∂∂∂∂⋅∂∂∂∂∂∂∂∂−=⋅−=∂∂∂∂−−imimmmmiyimixfxfyfyfyfyfffxyxyMLMLM11111111,,,,0),,;,,(0),,;,,(11111==mnmmnyyxxfyyxxfLLMLLB.2.3Ixpxp=+2211Ixp=⋅21xx−xIxpxpxp=++332211),,(321xxx),,(321pppn3Ixp=⋅n),,(1nxxL10C.Lagrange’sMethodcxGtsxfx=)(..)(maxxxxGfλ=λ))(()(),(xGcxfxL−+=λλxnλx0)(),(,,1,0),(=−=∂∂==−=∂∂xGcxLniGfxxLiiiλλλλLxxGfλ=xxGf,xn+1),,,(1λnxxLn+1),(λxλcxG=)(mλm),,(1mλλLxxGf⋅=λcxG=)(nm),,;,,(11mnxxλλLLn+mxxG),(λx11D.0)(..)(max≥≤xcxGtsxfx))(()(),(xGcxfxL−+=λλxnixLxxGfxLiiiiii,,1,0,0,0L==∂∂≥≤−=∂∂λxλλ0,0,0)(=∂∂≥≥−=∂∂λλλλLxGcLλbindingλnon-bindingx12Kuhn-TuckerConditionm(m1)λ,),(cxGm),,(21λxx0,021≥≥xx0,0;0,0;0,021===λxx8Quasi-LinearPreference21,maxxx1221ln),(xaxxxu+=..ts0,212211≥≤+xxIxpxp)(ln),,(22111221xpxpIxaxxxL−−++=λλ0,0,0111111=∂∂≥≤−=∂∂xLxxpxaxLλ0,0,0122222=∂∂≥≤−=∂∂xLxxpxLλ0,0,02211=∂∂≥≥−−=∂∂λλλλLxpxpIL0,0;0,0;0,021===λxx821,uu0λ01x0,0,021=λxx1300101122111=−=∂∂≤−=∂∂=−=∂∂xpILpxLpxaxLλλλ,,0,211IaxpIx===λ2apI≤0,0,021λxx0010221122111=−−=∂∂=−=∂∂=−=∂∂xpxpILpxLpxaxLλλλ22221211,,ppapIxpapx=−==λ2apI2apI≤0,0,021=λxx2apI0,0,021λxx121214E.E.1f(x)ABxE.1x0)('=xf)(xfx−)(xfABABE.1.1)(xf0xL+−+−+=200000))((''21))((')()(xxxfxxxfxfxf))(())((''21))((')()(20200000xxoxxxfxxxfxfxf−+−+−+=200000))((''21))((')()(xxxfxxxfxfxf−+−+≈)(xf0x15)(xfx2))((''21))((')()(xxxfxxxfxfxf−+−+≈)(xfx2))((''21)()(xxxfxfxf−≈−)(xfx0)(''xf0)()(−xfxf)()(xfxf)(xfx0)(''xf)(xfx0)(''xf)(xfxx)()(xfxf≥)(xfx0)(''≤xf0)(''≥xfxnx)()()(21)()()()(2xxoxxHxxxxxgradfxfxfT−+−⋅⋅−+−⋅+=xxx−n=nnnnffffH,,,,1111LMLnjifij,,1,,L=)(xfxHxx−Txx−xgradf)()(xfxf−16E.1.2AxxxQT=)(=nxxxM1xT),,(1nTxxxL=Ann×=nnnnaaaaA,,,,1111LMLnjiaajiij,,1,,L==0=xx)0(0)(≥xQ)(xQ0=xx)0(0)(≤xQ)(xQx)(xQAA)(xQA0,,0,021nDDDLA0)1(,,0,021−nnDDDLAk0,,0,021≥≥≥nDDDLAk0)1(,,0,021≥−≥≤nnDDDLnkaaaaaaDkkkkkk,,2,1,2111211LLML==knkniiaaaaaaDkiiiiiiiiiiiikkkkkk≤≤≤≤≤=,1,12112111LLMLH17H0)()(−xfxf)(xfxH)(xfxn=nnnnffffH,,,,1111LML0)1(,,0,011112221121111−nnnnnfffffffffLMLLH0,,0,011112221121111nnnnfffffffffLMLLn1E.2cxxgtsxxfxx=),(..),(max2121,21λ)),((),(),,(212121xxgcxxfxxL−+=λλx),,(21λxxLcxxg=),(21),,(21λxxLλ),(21xxf),,(21λxxLx180),(0021222111=−=∂∂=−=∂∂=−=∂∂xxgcLgfxLgfxLλλλ),(21xxfx2211dxfdxfdf+=21,ff),(21xxfx21,dxdxcxxg=),(2102211=+dxgdxg1212dxggdx−=21,ggdxdx1dx22dx121,,dxxxdx1),(21xxf)(2)()()()()()(222222211221112222222221121122122121112222222112112222111122221111221122112dxdfdxfdxdxfdxfdxxdxfdxfdxdxfdxxdxfdxdxfdxfdxxdxfdxfdxfdxxdxfdxfdxfdxxdxfdxfdxxdxfdxfdxfdxfddfdfd+++=∂∂+++∂∂++=∂∂+++∂∂++=∂+∂+∂+∂=+==Young2112ff=cxxg=),(210)(222222221122111=+++dxdgdxgdxdxgd