v2007.11.26 - Convex Optimization

∇ ∂g mn(X)∂X kl578 APPENDIX D. MATRIX CALCULUSx = a . A vector −υ based anywhere in domf × R pointing toward theunique bowl-bottom is specified:[ ] x − aυ ∝∈ R K × R (1612)f(x) + bSuch a vector issince the gradient isυ =⎡⎢⎣∇ x f(x)→∇ xf(x)1df(x)2⎤⎥⎦ (1613)∇ x f(x) = 2(x − a) (1614)and the directional derivative in the direction of the gradient is (1634)D.1.5→∇ xf(x)df(x) = ∇ x f(x) T ∇ x f(x) = 4(x − a) T (x − a) = 4(f(x) + b) (1615)Second directional derivativeBy similar argument, it so happens: the second directional derivative isequally simple. Given g(X) : R K×L →R M×N on open domain,⎡∂ 2 g mn(X)⎤∂X kl ∂X 12· · ·= ∂∇g mn(X)∂X kl=⎡∇ 2 g mn (X) =⎢⎣⎡=⎢⎣⎢⎣∇ ∂gmn(X)∂X 11∇ ∂gmn(X)∂X 21.∇ ∂gmn(X)∂X K1∂∇g mn(X)∂X 11∂∇g mn(X)∂X 21.∂∇g mn(X)∂X K1∂ 2 g mn(X)∂X kl ∂X 11∂ 2 g mn(X)∂X kl ∂X 21∂ 2 g mn(X)∂X kl ∂X K1∂ 2 g mn(X)∂X kl ∂X 1L∂ 2 g mn(X) ∂∂X kl ∂X 22· · ·2 g mn(X)∂X kl ∂X 2L∈ R K×L (1616)⎥. . . ⎦∂ 2 g mn(X) ∂∂X kl ∂X K2· · ·2 g mn(X)∂X kl ∂X KL⎤∇ ∂gmn(X)∂X 12· · · ∇ ∂gmn(X)∂X 1L∇ ∂gmn(X)∂X 22· · · ∇ ∂gmn(X)∂X 2L∈ R K×L×K×L⎥. . ⎦∇ ∂gmn(X)∂X K2· · · ∇ ∂gmn(X)∂X KL(1617)∂∇g mn(X)⎤∂X 12· · ·∂∇g mn(X)∂X 22.· · ·∂∇g mn(X)∂X K2· · ·∂∇g mn(X)∂X 1L∂∇g mn(X)∂X 2L.∂∇g mn(X)∂X KL⎥⎦