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1、AppendixDMatrixcalculusFromtoomuchstudy,andfromextremepassion,comethmadnesse.−IsaacNewton[150,§5]D.1Directionalderivative,TaylorseriesD.1.1GradientsKGradientofadifferentiablerealfunctionf(x):R→Rwithrespecttoitsvectorargumentisdefinedintermsofpartialderivatives∂f(
2、x)∂x1∂f(x)∇f(x),∂x2∈RK(1719)...∂f(x)∂xKwhilethesecond-ordergradientofthetwicedifferentiablerealfunctionwithrespecttoitsvectorargumentistraditionallycalledtheHessian;∂2f(x)∂2f(x)∂2f(x)2···∂x1∂x1∂x2∂x1∂xK∂2f(x)∂2f(x)∂2f(x)∇2f(x),∂x2∂x1∂x22···∂x2∂xK
3、∈SK(1720)............∂2f(x)∂2f(x)∂2f(x)···2∂xK∂x1∂xK∂x2∂xK©2001JonDattorro.co&edgversion2010.01.05.Allrightsreserved.657citation:Dattorro,ConvexOptimization&EuclideanDistanceGeometry,MεβooPublishingUSA,2005,v2010.01.05.658APPENDIXD.MATRIXCALCULUSNThegradien
4、tofvector-valuedfunctionv(x):R→Ronrealdomainisarow-vectorhi∇v(x),∂v1(x)∂v2(x)···∂vN(x)∈RN(1721)∂x∂x∂xwhilethesecond-ordergradientishi∇2v(x),∂2v1(x)∂2v2(x)∂2vN(x)∈RN(1722)∂x2∂x2···∂x2KNGradientofvector-valuedfunctionh(x):R→Ronvectordomainis∂h1(x)∂h2(x)···∂hN(x)∂
5、x1∂x1∂x1∂h1(x)∂h2(x)∂hN(x)···∇h(x),∂x2∂x2∂x2.........(1723)∂h1(x)∂h2(x)···∂hN(x)∂xK∂xK∂xKK×N=[∇h1(x)∇h2(x)···∇hN(x)]∈Rwhilethesecond-ordergradienthasathree-dimensionalrepresentationdubbedcubix;D.1∇∂h1(x)∇∂h2(x)···∇∂hN(x)∂x1∂x1∂x1∂h1(x)∂h2(x)∂hN(x
6、)∇2h(x),∇∂x2∇∂x2···∇∂x2.........(1724)∇∂h1(x)∇∂h2(x)···∇∂hN(x)∂xK∂xK∂xK=[∇2h(x)∇2h(x)···∇2h(x)]∈RK×N×K12Nwherethegradientofeachrealentryiswithrespecttovectorxasin(1719).D.1ThewordmatrixcomesfromtheLatinforwomb;relatedtotheprefixmatri-derivedfrommatermeani
7、ngmother.D.1.DIRECTIONALDERIVATIVE,TAYLORSERIES659K×LThegradientofrealfunctiong(X):R→Ronmatrixdomainis∂g(X)∂g(X)∂g(X)···∂X11∂X12∂X1L∂g(X)∂g(X)∂g(X)∇g(X),∂X21∂X22···∂X2L∈RK×L.........∂g(X)∂g(X)∂g(X)···∂XK1∂XK2∂XKL(1725)£∇X(:,1)g(X)∇X(:,2)g(X)K×1×L=∈R
8、...¤∇X(:,L)g(X)wherethegradient∇iswithrespecttotheithcolumnofX.TheX(:,i)K×1×Lstrangeappearanceof(1725)inRismeanttosuggestathirddimensionperpendiculartothepage(
