v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
684 APPENDIX D. MATRIX CALCULUSwhich can be proved by substitution of variables in (1821).second-order total differential due to any Y ∈R K×L isThe mn thd 2 g mn (X)| dX→Y= ∑ i,j∑k,l∂ 2 g mn (X)Y kl Y ij = tr(∇ X tr ( ∇g mn (X) T Y ) )TY∂X kl ∂X ij(1823)= ∑ ∂g mn (X + ∆t Y ) − ∂g mn (X)limY ij∆t→0 ∂Xi,jij ∆t(1824)g mn (X + 2∆t Y ) − 2g mn (X + ∆t Y ) + g mn (X)= lim∆t→0∆t 2 (1825)= d2dt 2 ∣∣∣∣t=0g mn (X+ t Y ) (1826)Hence the second directional derivative,→Ydg 2 (X) ⎡⎢⎣⎡tr(∇tr ( ∇g 11 (X) T Y ) )TY =tr(∇tr ( ∇g 21 (X) T Y ) )TY ⎢⎣ .tr(∇tr ( ∇g M1 (X) T Y ) )TYd 2 g 11 (X) d 2 g 12 (X) · · · d 2 g 1N (X)d 2 g 21 (X) d 2 g 22 (X) · · · d 2 g 2N (X). .d 2 g M1 (X) d 2 g M2 (X) · · ·.d 2 g MN (X)⎤⎥∈ R M×N⎦∣∣dX→Ytr(∇tr ( ∇g 12 (X) T Y ) )TY · · · tr(∇tr ( ∇g 1N (X) T Y ) ) ⎤TYtr(∇tr ( ∇g 22 (X) T Y ) )TY · · · tr(∇tr ( ∇g 2N (X) T Y ) )T Y ..tr(∇tr ( ∇g M2 (X) T Y ) )TY · · · tr(∇tr ( ∇g MN (X) T Y ) ⎥) ⎦TY⎡=⎢⎣∑ ∑i,jk,l∑ ∑i,jk,l∂ 2 g 11 (X)∂X kl ∂X ijY kl Y ij∂ 2 g 21 (X)∂X kl ∂X ijY kl Y ij.∑ ∑∂ 2 g M1 (X)∂X kl ∂X ijY kl Y iji,jk,l∑ ∑∂ 2 g 12 (X)∂X kl ∂X ijY kl Y ij · · ·i,ji,jk,l∑ ∑∂ 2 g 22 (X)∂X kl ∂X ijY kl Y ij · · ·k,l.∑ ∑∂ 2 g M2 (X)∂X kl ∂X ijY kl Y ij · · ·i,jk,l∑ ∑ ⎤∂ 2 g 1N (X)∂X kl ∂X ijY kl Y iji,j k,l∑ ∑∂ 2 g 2N (X)∂X kl ∂X ijY kl Y iji,j k,l. ⎥∑ ∑ ∂ 2 g MN (X) ⎦∂X kl ∂X ijY kl Y iji,jk,l(1827)
D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 685from which it follows→Ydg 2 (X) = ∑ i,j∑k,l∂ 2 g(X)∂X kl ∂X ijY kl Y ij = ∑ i,j∂∂X ij→Ydg(X)Y ij (1828)Yet for all X ∈ domg , any Y ∈R K×L , and some open interval of t∈Rg(X+ t Y ) = g(X) + t dg(X) →Y+ 1 →Yt2dg 2 (X) + o(t 3 ) (1829)2!which is the second-order Taylor series expansion about X . [219,18.4][152,2.3.4] Differentiating twice with respect to t and subsequent t-zeroingisolates the third term of the expansion. Thus differentiating and zeroingg(X+ t Y ) in t is an operation equivalent to individually differentiating andzeroing every entry g mn (X+ t Y ) as in (1826). So the second directionalderivative of g(X) : R K×L →R M×N becomes [277,2.1,5.4.5] [34,6.3.1]→Ydg 2 (X) = d2dt 2 ∣∣∣∣t=0g(X+ t Y ) ∈ R M×N (1830)which is again simplest. (confer (1809)) Directional derivative retains thedimensions of g .D.1.6directional derivative expressionsIn the case of a real function g(X) : R K×L →R , all its directional derivativesare in R :→Ydg(X) = tr ( ∇g(X) T Y ) (1831)→Ydg 2 (X) = tr(∇ X tr ( ∇g(X) T Y ) ) ( )T→YY = tr ∇ X dg(X) T Y(→Ydg 3 (X) = tr ∇ X tr(∇ X tr ( ∇g(X) T Y ) ) ) ( )T TY →YY = tr ∇ X dg 2 (X) T Y(1832)(1833)
- Page 633 and 634: Appendix BSimple matricesMathematic
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- Page 641 and 642: B.3. ELEMENTARY MATRIX 641has N −
- Page 643 and 644: B.4. AUXILIARY V -MATRICES 643is an
- Page 645 and 646: B.4. AUXILIARY V -MATRICES 64514. [
- Page 647 and 648: B.5. ORTHOGONAL MATRIX 647Given X
- Page 649 and 650: B.5. ORTHOGONAL MATRIX 649Figure 15
- Page 651 and 652: B.5. ORTHOGONAL MATRIX 651which is
- Page 653 and 654: Appendix CSome analytical optimal r
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- Page 661 and 662: C.3. ORTHOGONAL PROCRUSTES PROBLEM
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- Page 667 and 668: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
- Page 669 and 670: Appendix DMatrix calculusFrom too m
- Page 671 and 672: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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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
- Page 705 and 706: E.1. IDEMPOTENT MATRICES 705order,
- Page 707 and 708: E.1. IDEMPOTENT MATRICES 707are lin
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- Page 715 and 716: E.4. ALGEBRA OF PROJECTION ON AFFIN
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- Page 719 and 720: E.5. PROJECTION EXAMPLES 719where Y
- Page 721 and 722: E.5. PROJECTION EXAMPLES 721(B.4.2)
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- Page 725 and 726: E.6. VECTORIZATION INTERPRETATION,
- Page 727 and 728: E.6. VECTORIZATION INTERPRETATION,
- Page 729 and 730: E.7. PROJECTION ON MATRIX SUBSPACES
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- Page 733 and 734: E.8. RANGE/ROWSPACE INTERPRETATION
684 APPENDIX D. MATRIX CALCULUSwhich can be proved by substitution of variables in (1821).second-order total differential due to any Y ∈R K×L isThe mn thd 2 g mn (X)| dX→Y= ∑ i,j∑k,l∂ 2 g mn (X)Y kl Y ij = tr(∇ X tr ( ∇g mn (X) T Y ) )TY∂X kl ∂X ij(1823)= ∑ ∂g mn (X + ∆t Y ) − ∂g mn (X)limY ij∆t→0 ∂Xi,jij ∆t(1824)g mn (X + 2∆t Y ) − 2g mn (X + ∆t Y ) + g mn (X)= lim∆t→0∆t 2 (1825)= d2dt 2 ∣∣∣∣t=0g mn (X+ t Y ) (1826)Hence the second directional derivative,→Ydg 2 (X) ⎡⎢⎣⎡tr(∇tr ( ∇g 11 (X) T Y ) )TY =tr(∇tr ( ∇g 21 (X) T Y ) )TY ⎢⎣ .tr(∇tr ( ∇g M1 (X) T Y ) )TYd 2 g 11 (X) d 2 g 12 (X) · · · d 2 g 1N (X)d 2 g 21 (X) d 2 g 22 (X) · · · d 2 g 2N (X). .d 2 g M1 (X) d 2 g M2 (X) · · ·.d 2 g MN (X)⎤⎥∈ R M×N⎦∣∣dX→Ytr(∇tr ( ∇g 12 (X) T Y ) )TY · · · tr(∇tr ( ∇g 1N (X) T Y ) ) ⎤TYtr(∇tr ( ∇g 22 (X) T Y ) )TY · · · tr(∇tr ( ∇g 2N (X) T Y ) )T Y ..tr(∇tr ( ∇g M2 (X) T Y ) )TY · · · tr(∇tr ( ∇g MN (X) T Y ) ⎥) ⎦TY⎡=⎢⎣∑ ∑i,jk,l∑ ∑i,jk,l∂ 2 g 11 (X)∂X kl ∂X ijY kl Y ij∂ 2 g 21 (X)∂X kl ∂X ijY kl Y ij.∑ ∑∂ 2 g M1 (X)∂X kl ∂X ijY kl Y iji,jk,l∑ ∑∂ 2 g 12 (X)∂X kl ∂X ijY kl Y ij · · ·i,ji,jk,l∑ ∑∂ 2 g 22 (X)∂X kl ∂X ijY kl Y ij · · ·k,l.∑ ∑∂ 2 g M2 (X)∂X kl ∂X ijY kl Y ij · · ·i,jk,l∑ ∑ ⎤∂ 2 g 1N (X)∂X kl ∂X ijY kl Y iji,j k,l∑ ∑∂ 2 g 2N (X)∂X kl ∂X ijY kl Y iji,j k,l. ⎥∑ ∑ ∂ 2 g MN (X) ⎦∂X kl ∂X ijY kl Y iji,jk,l(1827)
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