v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
718 APPENDIX F. NOTATION AND A FEW DEFINITIONS int lim sgn round mod tr rank interior limit signum function or sign round to nearest integer modulus function matrix trace as in rankA, rank of matrix A ; dim R(A) dim dimension, dim R n = n , dim(x∈ R n ) = n , dim R(x∈ R n ) = 1, dim R(A∈ R m×n )= rank(A) aff dim aff affine hull affine dimension card cardinality, number of nonzero entries cardx ∆ = ‖x‖ 0 or N is cardinality of list X ∈ R n×N (p.287) conv cenv cone content cof convex hull convex envelope conic hull of high-dimensional bounded polyhedron, volume in 3 dimensions, area in 2, and so on matrix of cofactors corresponding to matrix argument dist distance between point or set arguments; e.g., dist(x, B) vec vectorization of m ×n matrix, Euclidean dimension mn (30) svec vectorization of symmetric n ×n matrix, Euclidean dimension n(n + 1)/2 (49) dvec vectorization of symmetric hollow n ×n matrix, Euclidean dimension n(n − 1)/2 (66)
(x,y) angle between vectors x and y , or dihedral angle between affine subsets 719 ≽ ≻ ⊁ ≥ generalized inequality; e.g., A ≽ 0 means vector or matrix A must be expressible in a biorthogonal expansion having nonnegative coordinates with respect to extreme directions of some implicit pointed closed convex cone K , or comparison to the origin with respect to some implicit pointed closed convex cone, or (when K= S+) n matrix A belongs to the positive semidefinite cone of symmetric matrices (2.9.0.1), or (when K= R n +) vector A belongs to the nonnegative orthant (each vector entry is nonnegative,2.3.1.1) strict generalized inequality, membership to cone interior not positive definite scalar inequality, greater than or equal to; comparison of scalars, or entrywise comparison of vectors or matrices with respect to R + nonnegative for α∈ R n , α ≽ 0 > greater than positive for α∈ R n , α ≻ 0 ; id est, no zero entries when with respect to nonnegative orthant, no vector on boundary with respect to pointed closed convex cone K ⌊ ⌋ floor function, ⌊x⌋ is greatest integer not exceeding x | | entrywise absolute value of scalars, vectors, and matrices log det natural (or Napierian) logarithm matrix determinant ‖x‖ vector 2-norm or Euclidean norm ‖x‖ 2 √ n∑ ‖x‖ l = l |x j | l vector l-norm for l ≥ 1 ∆ = j=1 n∑ |x j | l vector l-norm for 0 ≤ l < 1 j=1
- Page 667 and 668: E.6. VECTORIZATION INTERPRETATION,
- Page 669 and 670: E.7. ON VECTORIZED MATRICES OF HIGH
- Page 671 and 672: E.7. ON VECTORIZED MATRICES OF HIGH
- Page 673 and 674: E.8. RANGE/ROWSPACE INTERPRETATION
- Page 675 and 676: E.9. PROJECTION ON CONVEX SET 675 A
- Page 677 and 678: E.9. PROJECTION ON CONVEX SET 677 W
- Page 679 and 680: E.9. PROJECTION ON CONVEX SET 679 P
- Page 681 and 682: E.9. PROJECTION ON CONVEX SET 681 E
- Page 683 and 684: E.9. PROJECTION ON CONVEX SET 683 T
- Page 685 and 686: E.9. PROJECTION ON CONVEX SET 685
- Page 687 and 688: E.10. ALTERNATING PROJECTION 687 E.
- Page 689 and 690: E.10. ALTERNATING PROJECTION 689 b
- Page 691 and 692: E.10. ALTERNATING PROJECTION 691 a
- Page 693 and 694: E.10. ALTERNATING PROJECTION 693 (a
- Page 695 and 696: E.10. ALTERNATING PROJECTION 695 wh
- Page 697 and 698: E.10. ALTERNATING PROJECTION 697 E.
- Page 699 and 700: E.10. ALTERNATING PROJECTION 699 10
- Page 701 and 702: E.10. ALTERNATING PROJECTION 701 E.
- Page 703 and 704: E.10. ALTERNATING PROJECTION 703 E
- Page 705 and 706: Appendix F Notation and a few defin
- Page 707 and 708: 707 a.i. c.i. l.i. w.r.t affinely i
- Page 709 and 710: 709 is or ← → t → 0 + as in
- Page 711 and 712: 711 ∑ π(γ) Ξ Π ∏ ψ(Z) D D
- Page 713 and 714: 713 R m×n Euclidean vector space o
- Page 715 and 716: 715 H − H + ∂H ∂H ∂H −
- Page 717: 717 O O sort-index matrix order of
- Page 721 and 722: Bibliography [1] Suliman Al-Homidan
- Page 723 and 724: BIBLIOGRAPHY 723 [24] Alexander I.
- Page 725 and 726: BIBLIOGRAPHY 725 [52] Stephen Boyd,
- Page 727 and 728: BIBLIOGRAPHY 727 [78] Frank Critchl
- Page 729 and 730: BIBLIOGRAPHY 729 [105] Richard L. D
- Page 731 and 732: BIBLIOGRAPHY 731 [132] Michel X. Go
- Page 733 and 734: BIBLIOGRAPHY 733 [162] T. Herrmann,
- Page 735 and 736: BIBLIOGRAPHY 735 [191] Mark Kahrs a
- Page 737 and 738: BIBLIOGRAPHY 737 [220] K. V. Mardia
- Page 739 and 740: BIBLIOGRAPHY 739 [250] Pythagoras P
- Page 741 and 742: BIBLIOGRAPHY 741 [277] Anthony Man-
- Page 743 and 744: BIBLIOGRAPHY 743 [306] Michael W. T
- Page 745 and 746: [333] Margaret H. Wright. The inter
- Page 747 and 748: Index 0-norm, 203, 261, 294, 296, 2
- Page 749 and 750: INDEX 749 product, 43, 92, 147, 254
- Page 751 and 752: INDEX 751 coordinates, 140, 170, 17
- Page 753 and 754: INDEX 753 affine dimension, 485 fea
- Page 755 and 756: INDEX 755 affine, 209 nonlinear, 19
- Page 757 and 758: INDEX 757 of point, 37 ray, 90 rela
- Page 759 and 760: INDEX 759 normal, 47, 548, 563 norm
- Page 761 and 762: INDEX 761 strictly, 515, 520 functi
- Page 763 and 764: INDEX 763 vector, 45, 241, 248, 325
- Page 765 and 766: INDEX 765 convex envelope, see conv
- Page 767 and 768: INDEX 767 cone, 418, 420, 507 dual,
(x,y) angle between vectors x and y , or dihedral angle between affine<br />
subsets<br />
719<br />
≽<br />
≻<br />
⊁<br />
≥<br />
generalized inequality; e.g., A ≽ 0 means vector or matrix A must be<br />
expressible in a biorthogonal expansion having nonnegative coordinates<br />
with respect to extreme directions of some implicit pointed closed convex<br />
cone K , or comparison to the origin with respect to some implicit<br />
pointed closed convex cone, or (when K= S+) n matrix A belongs to the<br />
positive semidefinite cone of symmetric matrices (2.9.0.1), or (when<br />
K= R n +) vector A belongs to the nonnegative orthant (each vector entry<br />
is nonnegative,2.3.1.1)<br />
strict generalized inequality, membership to cone interior<br />
not positive definite<br />
scalar inequality, greater than or equal to; comparison of scalars,<br />
or entrywise comparison of vectors or matrices with respect to R +<br />
nonnegative for α∈ R n , α ≽ 0<br />
> greater than<br />
positive for α∈ R n , α ≻ 0 ; id est, no zero entries when with respect to<br />
nonnegative orthant, no vector on boundary with respect to pointed<br />
closed convex cone K<br />
⌊ ⌋ floor function, ⌊x⌋ is greatest integer not exceeding x<br />
| | entrywise absolute value of scalars, vectors, and matrices<br />
log<br />
det<br />
natural (or Napierian) logarithm<br />
matrix determinant<br />
‖x‖ vector 2-norm or Euclidean norm ‖x‖ 2<br />
√<br />
n∑<br />
‖x‖ l<br />
= l |x j | l vector l-norm for l ≥ 1<br />
∆<br />
=<br />
j=1<br />
n∑<br />
|x j | l vector l-norm for 0 ≤ l < 1<br />
j=1
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