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762 INDEX doubly stochastic matrices, 63 halfspace-description, 132 vertex-description, 59, 134 vertices, 134 polynomial, 558 constraint, 206, 315 objective, 315 quadratic, 558 polytope, 132 positive, 719 definite Farkas’ lemma, 256 semidefinite, 545 difference, 124 Farkas’ lemma, 255 square root, 564 strictly, 399 precision numerical, 248, 265, 291, 333, 525 primal feasible set, 254 problem, 146, 247 via dual, 261 principal component analysis, 323, 424 eigenvalue, 323 eigenvector, 323, 324 submatrix, 365, 435, 440, 464, 555 leading, 400, 402 theorem, 555 principle halfspace-description, 66 separating hyperplane, 77 supporting hyperplane, 75 problem 1-norm, 201–203, 262 nonnegative, 203, 297, 303, 304 Boolean, 63, 76, 261, 315, 409 cardinality, 296, 323, 333 Boolean, 261 nonnegative, 203, 294, 297, 303, 304 signed, 306 combinatorial, 204, 311, 315, 318, 536 completion, 348, 405, 412, 443, 452, 698, 699 compressed sensing, see compressed concave, 195, 246 conic, 146, 184, 246 convex, 146, 157, 184, 195, 200, 211, 218, 248, 255, 512, 673 geometry, 19, 25 nonlinear, 245 tractable, 246 dual, 144, 192, 204, 247, 259–261, 321 equivalent, 278 feasibility, 185 semidefinite, 251, 340 max cut, 318 minimax, 146 nonlinear, 365 convex, 245 prevalent, 501 primal, 146, 247 via dual, 261 Procrustes, see Procrustes proximity, 495, 497, 503 same, 278 statement as solution, 245 stress, 502, 523 procedure rank reduction, 269 Procrustes combinatorial, 311 compressed sensing, 314 diagonal, 608 linear program, 605 maximization, 605 orthogonal, 602 two sided, 604, 606 orthonormal, 310 solution unique, 603 symmetric, 607 translation, 603 product Cartesian, 43, 92, 147, 254, 695, 713 direct, 614 eigenvalue, 551, 552 Hadamard, 46, 514, 538, 615, 708 inner, 355
INDEX 763 vector, 45, 241, 248, 325, 379, 662, 709 vectorized matrix, 45, 547 Kronecker, 108, 540, 606, 614, 632 eigen, 540 inverse, 329 positive semidefinite, 352, 552 projector, 688 pseudoinverse, 641 vector, 616 vectorization, 539 outer, 59 positive definite nonsymmetric, 544 positive semidefinite symmetric, 555 pseudoinverse, 640 Kronecker, 641 tensor, 614 program, 76, 245 convex, see problem geometric, 8 integer, 63, 76, 409 linear, 76, 81, 82, 147, 199, 211, 245, 247, 248, 362 dual, 147, 247 prototypical, 147, 247 nonlinear convex, 245 semidefinite, 147, 199, 211, 245, 247, 313 dual, 147, 247 prototypical, 147, 247 Schur-form, 206, 219, 516, 527, 682 projection, 47, 639 algebra, 654 alternating, 361, 687, 688 convergence, 692, 694, 696, 699 distance, 690, 691 Dykstra algorithm, 687, 700, 701 feasibility, 690–692 iteration, 687, 701 on affine/psd cone, 697 on EDM cone, 527 on halfspaces, 689 on orthant/hyperplane, 693, 694 optimization, 690, 691, 700 over/under, 695 biorthogonal expansion, 656 coefficient, 167, 472, 661, 668 cyclic, 689 direction, 649, 652, 664 nonorthogonal, 646, 647, 661 dual, 674 on cone, 677 on convex set, 674, 676 on EDM cone, 529 optimization, 675 easy, 683 Euclidean, 122, 497, 500, 504, 506, 514, 525, 674 matrix, see matrix minimum-distance, 647, 650, 670, 673 nonorthogonal, 513, 646, 647, 661 eigenvalue coefficients, 663 oblique, 201, 202, 502, 643 of matrix, 669, 671 of origin, 201, 202, 643, 659 of PSD cone, 110 of range/rowspace, 673 on affine, 44, 655, 659, 661 origin, 201, 202, 643, 659 vertex-description, 660 on algebraic complement, 648 on basis nonorthogonal, 655 orthogonal, 655 on box, 683 on cardinality k , 684 on cone, 678, 680, 684 EDM, 515, 525 polyhedral, 684 truncated, 681 on convex set, 673 in affine subset, 686 in subspace, 685, 686 on convex sets, 689 on dyad, 662, 664 on ellipsoid, 308 on Euclidean ball, 683
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DATTORRO CONVEX OPTIMIZATION & EUCL
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Convex Optimization & Euclidean Dis
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for Jennie Columba ♦ Antonio ♦
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Prelude The constant demands of my
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Convex Optimization & Euclidean Dis
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures 1 Overview 19 1 Ori
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LIST OF FIGURES 15 3 Geometry of co
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LIST OF FIGURES 17 126 Decomposing
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Chapter 1 Overview Convex Optimizat
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ˇx 4 ˇx 3 ˇx 2 Figure 2: Applica
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23 Figure 4: This coarsely discreti
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ases (biorthogonal expansion). We e
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27 Figure 7: These bees construct a
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that establish its membership to th
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31 appendices Provided so as to be
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Chapter 2 Convex geometry Convexity
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2.1. CONVEX SET 35 2.1.2 linear ind
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2.1. CONVEX SET 37 2.1.6 empty set
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2.1. CONVEX SET 39 2.1.7.1 Line int
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2.1. CONVEX SET 41 (a) R 2 (b) R 3
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2.1. CONVEX SET 43 This theorem in
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 55 Figure 16: Convex hul
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2.3. HULLS 57 The affine hull of tw
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2.3. HULLS 59 2.3.2 Convex hull The
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2.3. HULLS 61 In case k = N , the F
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2.3. HULLS 63 2.3.2.0.3 Exercise. C
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2.3. HULLS 65 Figure 20: A simplici
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2.4. HALFSPACE, HYPERPLANE 67 H + a
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2.4. HALFSPACE, HYPERPLANE 69 1 1
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2.4. HALFSPACE, HYPERPLANE 71 Recal
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2.4. HALFSPACE, HYPERPLANE 73 C H
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2.4. HALFSPACE, HYPERPLANE 75 2.4.2
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2.4. HALFSPACE, HYPERPLANE 77 (conf
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2.5. SUBSPACE REPRESENTATIONS 79 Ra
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2.5. SUBSPACE REPRESENTATIONS 81 If
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2.6. EXTREME, EXPOSED 83 2.6 Extrem
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2.6. EXTREME, EXPOSED 85 A B C D Fi
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2.7. CONES 87 2.6.1.3.1 Definition.
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2.7. CONES 89 0 Figure 30: Boundary
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2.7. CONES 91 2.7.2 Convex cone We
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2.7. CONES 93 Then a pointed closed
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2.7. CONES 95 A pointed closed conv
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2.8. CONE BOUNDARY 97 So the ray th
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2.8. CONE BOUNDARY 99 2.8.1.1 extre
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2.8. CONE BOUNDARY 101 2.8.2 Expose
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2.8. CONE BOUNDARY 103 From Theorem
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 127
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2.10. CONIC INDEPENDENCE (C.I.) 129
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2.10. CONIC INDEPENDENCE (C.I.) 131
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2.12. CONVEX POLYHEDRA 133 It follo
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2.12. CONVEX POLYHEDRA 135 Coeffici
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2.12. CONVEX POLYHEDRA 137 2.12.3 U
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2.12. CONVEX POLYHEDRA 139
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2.13. DUAL CONE & GENERALIZED INEQU
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2.13. DUAL CONE & GENERALIZED INEQU
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3 Geometry of convex functi
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3.1. CONVEX FUNCTION 197 f 1 (x) f
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3.1. CONVEX FUNCTION 199 3.1.3 norm
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3.1. CONVEX FUNCTION 201 A B 1 Figu
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3.1. CONVEX FUNCTION 203 k/m 1 0.9
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k∑ i=1 3.1. CONVEX FUNCTION 205 S
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3.1. CONVEX FUNCTION 207 rather x >
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3.1. CONVEX FUNCTION 209 rather ] x
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3.1. CONVEX FUNCTION 211 3.1.6.0.2
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3.1. CONVEX FUNCTION 213 q(x) f(x)
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3.1. CONVEX FUNCTION 215 3.1.7.0.2
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3.1. CONVEX FUNCTION 217 We learned
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3.1. CONVEX FUNCTION 219 Since opti
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3.1. CONVEX FUNCTION 221 2 1.5 1 0.
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3.1. CONVEX FUNCTION 223 Setting th
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3.1. CONVEX FUNCTION 225 Similarly,
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3.1. CONVEX FUNCTION 227 For vector
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3.1. CONVEX FUNCTION 229 This means
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3.1. CONVEX FUNCTION 231 f(Y ) −
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 239 exponential al
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3.3. QUASICONVEX 241 Unlike convex
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3.4. SALIENT PROPERTIES 243 6. (af
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Chapter 4 Semidefinite programming
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4.1. CONIC PROBLEM 247 where K is a
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4.1. CONIC PROBLEM 249 4.1.1.2 Redu
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4.1. CONIC PROBLEM 251 In any SDP f
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4.1. CONIC PROBLEM 253 Proposition
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4.2. FRAMEWORK 255 sets are closed
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4.2. FRAMEWORK 257 4.2.1.1.3 Exampl
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4.2. FRAMEWORK 259 4.2.2 Duals The
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4.2. FRAMEWORK 261 When equality is
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4.2. FRAMEWORK 263 The pseudoinvers
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4.2. FRAMEWORK 265 For the data giv
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4.2. FRAMEWORK 267 minimizes an aff
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4.3. RANK REDUCTION 269 whose rank
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4.3. RANK REDUCTION 271 and where m
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4.3. RANK REDUCTION 273 4.3.3 Optim
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4.3. RANK REDUCTION 275 Initialize:
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 295 m
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4.5. CONSTRAINING CARDINALITY 297 m
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4.5. CONSTRAINING CARDINALITY 299 a
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4.5. CONSTRAINING CARDINALITY 301 f
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4.5. CONSTRAINING CARDINALITY 303 n
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4.5. CONSTRAINING CARDINALITY 305 W
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4.5. CONSTRAINING CARDINALITY 307 t
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.7. CONVEX ITERATION RANK-1 341 fi
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4.7. CONVEX ITERATION RANK-1 343 Gi
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Chapter 5 Euclidean Distance Matrix
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5.2. FIRST METRIC PROPERTIES 347 co
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
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5.4. EDM DEFINITION 353 The collect
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5.4. EDM DEFINITION 355 5.4.2 Gram-
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5.4. EDM DEFINITION 357 D ∈ EDM N
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5.4. EDM DEFINITION 359 5.4.2.2.1 E
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5.4. EDM DEFINITION 361 ten affine
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5.4. EDM DEFINITION 363 spheres: Th
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5.4. EDM DEFINITION 365 By eliminat
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5.4. EDM DEFINITION 367 where Φ ij
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5.4. EDM DEFINITION 369 5.4.2.2.6 D
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5.4. EDM DEFINITION 371 10 5 ˇx 4
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5.4. EDM DEFINITION 373 corrected b
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5.4. EDM DEFINITION 375 by translat
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5.4. EDM DEFINITION 377 Crippen & H
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5.4. EDM DEFINITION 379 where ([√
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5.4. EDM DEFINITION 381 because (A.
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5.5. INVARIANCE 383 5.5.1.0.1 Examp
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5.5. INVARIANCE 385 x 2 x 2 x 3 x 1
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 393 5
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5.7. EMBEDDING IN AFFINE HULL 395 F
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5.7. EMBEDDING IN AFFINE HULL 397 5
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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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5.10. EDM-ENTRY COMPOSITION 413 of
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5.10. EDM-ENTRY COMPOSITION 415 The
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5.11. EDM INDEFINITENESS 417 5.11.1
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5.11. EDM INDEFINITENESS 419 we hav
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5.11. EDM INDEFINITENESS 421 So bec
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5.11. EDM INDEFINITENESS 423 where
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5.12. LIST RECONSTRUCTION 425 where
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5.12. LIST RECONSTRUCTION 427 (a) (
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5.13. RECONSTRUCTION EXAMPLES 429 D
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5.13. RECONSTRUCTION EXAMPLES 431 T
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5.13. RECONSTRUCTION EXAMPLES 433 w
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6 Cone of distance matrices
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6.1. DEFINING EDM CONE 447 6.1 Defi
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6.2. POLYHEDRAL BOUNDS 449 This con
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6.3. √ EDM CONE IS NOT CONVEX 451
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6.4. A GEOMETRY OF COMPLETION 453 (
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6.4. A GEOMETRY OF COMPLETION 455 (
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6.4. A GEOMETRY OF COMPLETION 457 F
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6.5. EDM DEFINITION IN 11 T 459 by
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6.5. EDM DEFINITION IN 11 T 461 6.5
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6.5. EDM DEFINITION IN 11 T 463 1 0
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6.5. EDM DEFINITION IN 11 T 465 6.5
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6.6. CORRESPONDENCE TO PSD CONE S N
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6.6. CORRESPONDENCE TO PSD CONE S N
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6.6. CORRESPONDENCE TO PSD CONE S N
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6.7. VECTORIZATION & PROJECTION INT
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6.7. VECTORIZATION & PROJECTION INT
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6.8. DUAL EDM CONE 477 When the Fin
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6.8. DUAL EDM CONE 479 Proof. First
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6.8. DUAL EDM CONE 481 EDM 2 = S 2
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6.8. DUAL EDM CONE 483 whose veraci
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6.8. DUAL EDM CONE 485 6.8.1.3.1 Ex
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6.8. DUAL EDM CONE 487 has dual aff
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6.8. DUAL EDM CONE 489 6.8.1.7 Scho
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6.9. THEOREM OF THE ALTERNATIVE 491
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6.10. POSTSCRIPT 493 When D is an E
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Chapter 7 Proximity problems In sum
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497 project on the subspace, then p
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499 H S N h 0 EDM N K = S N h ∩ R
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501 7.0.3 Problem approach Problems
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7.1. FIRST PREVALENT PROBLEM: 503 f
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7.1. FIRST PREVALENT PROBLEM: 505 7
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7.1. FIRST PREVALENT PROBLEM: 507 d
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7.1. FIRST PREVALENT PROBLEM: 509 7
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7.1. FIRST PREVALENT PROBLEM: 511 w
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7.1. FIRST PREVALENT PROBLEM: 513 T
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7.2. SECOND PREVALENT PROBLEM: 515
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7.2. SECOND PREVALENT PROBLEM: 517
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7.2. SECOND PREVALENT PROBLEM: 519
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7.2. SECOND PREVALENT PROBLEM: 521
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7.2. SECOND PREVALENT PROBLEM: 523
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7.3. THIRD PREVALENT PROBLEM: 525 g
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7.3. THIRD PREVALENT PROBLEM: 527 w
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7.3. THIRD PREVALENT PROBLEM: 529 7
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Page 531 and 532:
7.3. THIRD PREVALENT PROBLEM: 531 7
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7.3. THIRD PREVALENT PROBLEM: 533 O
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7.4. CONCLUSION 535 The rank constr
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Page 537 and 538:
Appendix A Linear algebra A.1 Main-
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Page 539 and 540:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
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Page 541 and 542:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
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Page 543 and 544:
A.2. SEMIDEFINITENESS: DOMAIN OF TE
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Page 545 and 546:
A.3. PROPER STATEMENTS 545 (AB) T
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Page 547 and 548:
A.3. PROPER STATEMENTS 547 A.3.1 Se
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Page 549 and 550:
A.3. PROPER STATEMENTS 549 For A di
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Page 551 and 552:
A.3. PROPER STATEMENTS 551 Diagonal
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Page 553 and 554:
A.3. PROPER STATEMENTS 553 For A,B
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Page 555 and 556:
A.3. PROPER STATEMENTS 555 A.3.1.0.
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A.4. SCHUR COMPLEMENT 557 A.4 Schur
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A.4. SCHUR COMPLEMENT 559 A.4.0.0.2
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Page 561 and 562:
A.5. EIGEN DECOMPOSITION 561 When B
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Page 563 and 564:
A.5. EIGEN DECOMPOSITION 563 dim N(
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Page 565 and 566:
A.6. SINGULAR VALUE DECOMPOSITION,
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.7. ZEROS 569 Given symmetric matr
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Page 571 and 572:
A.7. ZEROS 571 (TRANSPOSE.) Likewis
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A.7. ZEROS 573 For X,A∈ S M + [31
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A.7. ZEROS 575 A.7.5.0.1 Propositio
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Appendix B Simple matrices Mathemat
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Page 579 and 580:
B.1. RANK-ONE MATRIX (DYAD) 579 R(v
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Page 581 and 582:
B.1. RANK-ONE MATRIX (DYAD) 581 ran
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Page 583 and 584:
B.2. DOUBLET 583 R([u v ]) R(Π)= R
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Page 585 and 586:
B.3. ELEMENTARY MATRIX 585 If λ
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Page 587 and 588:
B.4. AUXILIARY V -MATRICES 587 the
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Page 589 and 590:
B.4. AUXILIARY V -MATRICES 589 18.
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Page 591 and 592:
B.5. ORTHOGONAL MATRIX 591 B.5 Orth
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B.5. ORTHOGONAL MATRIX 593 Figure 1
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Page 595 and 596:
Appendix C Some analytical optimal
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Page 597 and 598:
C.2. TRACE, SINGULAR AND EIGEN VALU
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Page 599 and 600:
C.2. TRACE, SINGULAR AND EIGEN VALU
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Page 601 and 602:
C.2. TRACE, SINGULAR AND EIGEN VALU
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Page 603 and 604:
C.3. ORTHOGONAL PROCRUSTES PROBLEM
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Page 605 and 606:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Page 607 and 608:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Page 609 and 610:
Appendix D Matrix calculus From too
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Page 611 and 612:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 613 and 614:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 615 and 616:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 617 and 618:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 619 and 620:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 621 and 622:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 623 and 624:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 625 and 626:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 627 and 628:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 629 and 630:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 631 and 632:
D.2. TABLES OF GRADIENTS AND DERIVA
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Page 633 and 634:
D.2. TABLES OF GRADIENTS AND DERIVA
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Page 635 and 636:
D.2. TABLES OF GRADIENTS AND DERIVA
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Page 637 and 638:
D.2. TABLES OF GRADIENTS AND DERIVA
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Page 639 and 640:
Appendix E Projection For any A∈
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Page 641 and 642:
641 U T = U † for orthonormal (in
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Page 643 and 644:
E.1. IDEMPOTENT MATRICES 643 where
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Page 645 and 646:
E.1. IDEMPOTENT MATRICES 645 order,
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Page 647 and 648:
E.1. IDEMPOTENT MATRICES 647 When t
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Page 649 and 650:
E.3. SYMMETRIC IDEMPOTENT MATRICES
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Page 651 and 652:
E.3. SYMMETRIC IDEMPOTENT MATRICES
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Page 653 and 654:
E.3. SYMMETRIC IDEMPOTENT MATRICES
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Page 655 and 656:
E.5. PROJECTION EXAMPLES 655 E.4.0.
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Page 657 and 658:
E.5. PROJECTION EXAMPLES 657 a ∗
-
Page 659 and 660:
E.5. PROJECTION EXAMPLES 659 E.5.0.
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Page 661 and 662:
E.6. VECTORIZATION INTERPRETATION,
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Page 663 and 664:
E.6. VECTORIZATION INTERPRETATION,
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Page 665 and 666:
E.6. VECTORIZATION INTERPRETATION,
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Page 667 and 668:
E.6. VECTORIZATION INTERPRETATION,
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Page 669 and 670:
E.7. ON VECTORIZED MATRICES OF HIGH
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Page 671 and 672:
E.7. ON VECTORIZED MATRICES OF HIGH
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Page 673 and 674:
E.8. RANGE/ROWSPACE INTERPRETATION
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Page 675 and 676:
E.9. PROJECTION ON CONVEX SET 675 A
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Page 677 and 678:
E.9. PROJECTION ON CONVEX SET 677 W
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Page 679 and 680:
E.9. PROJECTION ON CONVEX SET 679 P
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Page 681 and 682:
E.9. PROJECTION ON CONVEX SET 681 E
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Page 683 and 684:
E.9. PROJECTION ON CONVEX SET 683 T
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Page 685 and 686:
E.9. PROJECTION ON CONVEX SET 685
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Page 687 and 688:
E.10. ALTERNATING PROJECTION 687 E.
-
Page 689 and 690:
E.10. ALTERNATING PROJECTION 689 b
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Page 691 and 692:
E.10. ALTERNATING PROJECTION 691 a
-
Page 693 and 694:
E.10. ALTERNATING PROJECTION 693 (a
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Page 695 and 696:
E.10. ALTERNATING PROJECTION 695 wh
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Page 697 and 698:
E.10. ALTERNATING PROJECTION 697 E.
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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
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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
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Page 715 and 716:
715 H − H + ∂H ∂H ∂H −
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Page 717 and 718:
717 O O sort-index matrix order of
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Page 719 and 720:
(x,y) angle between vectors x and y
-
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,
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Page 727 and 728:
BIBLIOGRAPHY 727 [78] Frank Critchl
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Page 729 and 730:
BIBLIOGRAPHY 729 [105] Richard L. D
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Page 731 and 732:
BIBLIOGRAPHY 731 [132] Michel X. Go
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Page 733 and 734:
BIBLIOGRAPHY 733 [162] T. Herrmann,
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Page 735 and 736:
BIBLIOGRAPHY 735 [191] Mark Kahrs a
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Page 737 and 738:
BIBLIOGRAPHY 737 [220] K. V. Mardia
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Page 739 and 740:
BIBLIOGRAPHY 739 [250] Pythagoras P
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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
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Page 747 and 748:
Index 0-norm, 203, 261, 294, 296, 2
-
Page 749 and 750:
INDEX 749 product, 43, 92, 147, 254
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Page 751 and 752:
INDEX 751 coordinates, 140, 170, 17
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Page 753 and 754:
INDEX 753 affine dimension, 485 fea
-
Page 755 and 756:
INDEX 755 affine, 209 nonlinear, 19
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Page 757 and 758:
INDEX 757 of point, 37 ray, 90 rela
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Page 759 and 760:
INDEX 759 normal, 47, 548, 563 norm
-
Page 761:
INDEX 761 strictly, 515, 520 functi
-
Page 765 and 766:
INDEX 765 convex envelope, see conv
-
Page 767 and 768:
INDEX 767 cone, 418, 420, 507 dual,
-
Page 769 and 770:
INDEX 769 trilateration, 21, 42, 36
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Page 772:
Convex Optimization & Euclidean Dis
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