v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
∇ ∂g mn(X)∂X kl682 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 (1812)f(x) + bSuch a vector issince the gradient isυ =⎡⎢⎣∇ x f(x)→∇ xf(x)1df(x)2⎤⎥⎦ (1813)∇ x f(x) = 2(x − a) (1814)and the directional derivative in the direction of the gradient is (1834)D.1.5→∇ xf(x)df(x) = ∇ x f(x) T ∇ x f(x) = 4(x − a) T (x − a) = 4(f(x) + b) (1815)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 22· · ·.∂ 2 g mn(X)∂X kl ∂X K2· · ·∇ ∂gmn(X)∂X 12· · · ∇ ∂gmn(X)∂X 1L∇ ∂gmn(X)∂X 22· · · ∇ ∂gmn(X)∂X 2L∂ 2 g mn(X)∂X kl ∂X 1L∂ 2 g mn(X)∂X kl ∂X 2L.∂ 2 g mn(X)∂X kl ∂X KL∈ R K×L (1816)⎥⎦∈ R K×L×K×L⎥. . ⎦∇ ∂gmn(X)∂X K2· · · ∇ ∂gmn(X)∂X KL∂∇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⎤⎥⎦⎤(1817)
D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 683Rotating our perspective, we get several views of the second-order gradient:⎡∇ 2 g(X) = ⎢⎣∇ 2 g 11 (X) ∇ 2 g 12 (X) · · · ∇ 2 g 1N (X)∇ 2 g 21 (X) ∇ 2 g 22 (X) · · · ∇ 2 g 2N (X). .∇ 2 g M1 (X) ∇ 2 g M2 (X) · · ·.∇ 2 g MN (X)⎤⎥⎦ ∈ RM×N×K×L×K×L (1818)⎡∇ 2 g(X) T 1=⎢⎣∇ ∂g(X)∂X 11∇ ∂g(X)∂X 21.∇ ∂g(X)∂X K1∇ ∂g(X)∂X 12· · · ∇ ∂g(X)∂X 1L∇ ∂g(X)∂X 22.· · · ∇ ∂g(X)∂X 2L.∇ ∂g(X)∂X K2· · · ∇ ∂g(X)∂X KL⎤⎥⎦ ∈ RK×L×M×N×K×L (1819)⎡∇ 2 g(X) T 2=⎢⎣∂∇g(X)∂X 11∂∇g(X)∂X 21.∂∇g(X)∂X K1∂∇g(X)∂X 12· · ·∂∇g(X)∂X 22.· · ·∂∇g(X)∂X K2· · ·∂∇g(X)∂X 1L∂∇g(X)∂X 2L.∂∇g(X)∂X KL⎤⎥⎦ ∈ RK×L×K×L×M×N (1820)Assuming the limits exist, we may state the partial derivative of the mn thentry of g with respect to the kl th and ij th entries of X ;∂ 2 g mn(X)∂X kl ∂X ij=g mn(X+∆t elimk e T l +∆τ e i eT j )−gmn(X+∆t e k eT l )−(gmn(X+∆τ e i eT)−gmn(X))j∆τ,∆t→0∆τ ∆t(1821)Differentiating (1801) and then scaling by Y ij∂ 2 g mn(X)∂X kl ∂X ijY kl Y ij = lim∆t→0∂g mn(X+∆t Y kl e k e T l )−∂gmn(X)∂X ijY∆tij(1822)g mn(X+∆t Y kl e= limk e T l +∆τ Y ij e i e T j )−gmn(X+∆t Y kl e k e T l )−(gmn(X+∆τ Y ij e i e T)−gmn(X))j∆τ,∆t→0∆τ ∆t
- Page 631 and 632: A.7. ZEROS 631A.7.5.0.1 Proposition
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- Page 669 and 670: Appendix DMatrix calculusFrom too m
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- Page 699 and 700: Appendix EProjectionFor any A∈ R
- Page 701 and 702: 701U T = U † for orthonormal (inc
- Page 703 and 704: E.1. IDEMPOTENT MATRICES 703where A
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- Page 707 and 708: E.1. IDEMPOTENT MATRICES 707are lin
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D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 683Rotating our perspective, we get several views of the second-order gradient:⎡∇ 2 g(X) = ⎢⎣∇ 2 g 11 (X) ∇ 2 g 12 (X) · · · ∇ 2 g 1N (X)∇ 2 g 21 (X) ∇ 2 g 22 (X) · · · ∇ 2 g 2N (X). .∇ 2 g M1 (X) ∇ 2 g M2 (X) · · ·.∇ 2 g MN (X)⎤⎥⎦ ∈ RM×N×K×L×K×L (1818)⎡∇ 2 g(X) T 1=⎢⎣∇ ∂g(X)∂X 11∇ ∂g(X)∂X 21.∇ ∂g(X)∂X K1∇ ∂g(X)∂X 12· · · ∇ ∂g(X)∂X 1L∇ ∂g(X)∂X 22.· · · ∇ ∂g(X)∂X 2L.∇ ∂g(X)∂X K2· · · ∇ ∂g(X)∂X KL⎤⎥⎦ ∈ RK×L×M×N×K×L (1819)⎡∇ 2 g(X) T 2=⎢⎣∂∇g(X)∂X 11∂∇g(X)∂X 21.∂∇g(X)∂X K1∂∇g(X)∂X 12· · ·∂∇g(X)∂X 22.· · ·∂∇g(X)∂X K2· · ·∂∇g(X)∂X 1L∂∇g(X)∂X 2L.∂∇g(X)∂X KL⎤⎥⎦ ∈ RK×L×K×L×M×N (1820)Assuming the limits exist, we may state the partial derivative of the mn thentry of g with respect to the kl th and ij th entries of X ;∂ 2 g mn(X)∂X kl ∂X ij=g mn(X+∆t elimk e T l +∆τ e i eT j )−gmn(X+∆t e k eT l )−(gmn(X+∆τ e i eT)−gmn(X))j∆τ,∆t→0∆τ ∆t(1821)Differentiating (1801) and then scaling by Y ij∂ 2 g mn(X)∂X kl ∂X ijY kl Y ij = lim∆t→0∂g mn(X+∆t Y kl e k e T l )−∂gmn(X)∂X ijY∆tij(1822)g mn(X+∆t Y kl e= limk e T l +∆τ Y ij e i e T j )−gmn(X+∆t Y kl e k e T l )−(gmn(X+∆τ Y ij e i e T)−gmn(X))j∆τ,∆t→0∆τ ∆t
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