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1、AppendixDMatrixcalculusFromtoomuchstudy,andfromextremepassion,comethmadnesse.−IsaacNewton[179,§5]D.1Gradient,Directionalderivative,TaylorseriesD.1.1GradientsKGradientofadifferentiablerealfunctionf(x):R→Rwithrespecttoitsvectorargumentisdefineduniquelyintermsofpartialderivatives∂f(x)∂x1∂f(
2、x)∂xK∇f(x),.2∈R(1955)..∂f(x)∂xKwhilethesecond-ordergradientofthetwicedifferentiablerealfunctionwithrespecttoitsvectorargumentistraditionallycalledtheHessian;222∂f(x)∂f(x)∂f(x)∂x21∂x1∂x2···∂x1∂xK222∂f(x)∂f(x)···∂f(x)2∂x2∂x1∂x2∂x2∂xKK∇f(x),2∈S(1956)............222∂f(x)
3、∂f(x)∂f(x)∂xK∂x1∂xK∂x2···∂xK2NThegradientofvector-valuedfunctionv(x):R→Ronrealdomainisarowvectorhi∇v(x),∂v1(x)∂v2(x)···∂vN(x)∈RN(1957)∂x∂x∂xwhilethesecond-ordergradientishi2∂2v(x)∂2v(x)∂2v(x)N∇v(x),12···N∈R(1958)∂x2∂x2∂x2Dattorro,ConvexOptimizationEuclideanDistanceGeometry2ε,Mεβoo,v2018.09.
4、21.549550APPENDIXD.MATRIXCALCULUSKNGradientofvector-valuedfunctionh(x):R→Ronvectordomainis∂h1(x)∂h2(x)···∂hN(x)∂x1∂x1∂x1∂h1(x)∂h2(x)···∂hN(x)∇h(x),∂x2∂x2∂x2.........(1959)∂h1(x)∂h2(x)···∂hN(x)∂xK∂xK∂xKK×N=[∇h1(x)∇h2(x)···∇hN(x)]∈Rwhilethesecond-ordergradienthasathree-dimension
5、alwrittenrepresentationdubbedcubix;D.1∇∂h1(x)∇∂h2(x)···∇∂hN(x)∂x1∂x1∂x1∇∂h1(x)∇∂h2(x)···∇∂hN(x)∇2h(x),∂x2∂x2∂x2.........(1960)∇∂h1(x)∇∂h2(x)···∇∂hN(x)∂xK∂xK∂xK£¤=∇2h(x)∇2h(x)···∇2h(x)∈RK×N×K12Nwherethegradientofeachrealentryiswithrespecttovectorxasin(1955).K×LThegradientofreal
6、functiong(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(1961)£∇X(:,1)g(X)∇X(:,2)g(X)K×1×L=∈R...¤∇X(:,L)g(X)wheregradient∇iswithrespecttotheithcolumnofX.ThestrangeappearanceofX(:,i)K×1×L(1961
7、)inRismeanttosuggestathirddimensionperpendiculartothepage(notadiagonalmatrix).Thesecond-ordergradienthasrepresentationD.1ThewordmatrixcomesfromtheLatinforwomb;relatedtotheprefixmatri-derivedfrommatermeaningmother.D.1.GRADIENT,DIRECTIONALDERIVATIVE,TAYLORS
