10.03.2015
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748 INDEX interpoint, 355 obtuse, 66 positive semidefinite cone, 114 right, 115, 143, 161, 162 torsion, 27, 376 antidiagonal, 385 antisymmetric, 50 antihollow, 53, 497 antisymmetry, 93, 552 artificial intelligence, 23 axioms of the metric, 347 axis of revolution, 115, 116, 592 ball 1-norm, 201, 202 nonnegative, 305 ∞-norm, 201 Euclidean, 59, 202, 361, 643, 683 packing, 361 Barvinok, 33, 104, 124, 127, 130, 253, 344 base, 87 station, 373 basis, 52, 170, 712 discrete cosine transform, 299 nonorthogonal, 170, 655 projection on, 655 orthogonal projection, 655 orthonormal, 52, 81, 652 pursuit, 326 standard, 52, 55, 137 bees, 27, 361 bijection, see bijective bijective, 45, 109, 606 Gram form, 387 inner-product form, 391 isometry, 48, 433 linear, 47, 48, 52, 54, 129, 164, 511, 591 pseudoinverse, 640 biorthogonal decomposition, 644, 652 expansion, 140, 165, 170, 645 EDM, 465 projection, 656 unique, 167, 170, 172, 466, 582, 657 biorthogonality, 168, 170 condition, 167, 563, 582 blunt, 704 bound lower, 500 polyhedral, 449 boundary, 38, 86, 469 classical, 38 cone, 96 EDM, 447, 462 membership, 153, 258 PSD, 104, 109, 121, 474 ray, 97 conventional, 87 extreme, 101 halfspace, 66 of point, 87 of ray, 90 open set, 38 point, 85 hypercube slice, 336 relative, 37, 59, 86, 87 bounded, 59 bowl, 620, 621 box, 76, 683 bridge, 377, 406, 456, 501, 517 calculus matrix, 609 of inequalities, 7 of variations, 157 Candès, 325 Carathéodory’s theorem, 152, 666 cardinality, 352 -1, 305 convex envelope, see convex geometry, 201, 305 minimization, 201, 202, 296, 323, 333 Boolean, 261 nonnegative, 203, 294, 297, 303, 304 signed, 306 problem, see cardinality minimization regularization, 300, 323 Cartesian axes, 36, 58, 73, 90, 98, 138, 150 cone, 140, 152 coordinates, 428
INDEX 749 product, 43, 92, 147, 254, 695, 713 Cayley-Menger determinant, 439, 441 form, 397, 418, 419, 507, 531 cellular, 374 telephone network, 373 center geometric, 357, 383, 426 gravity, 383 mass, 383 central path, 254 certificate null, 154 chain rule, 616 two argument, 616 Chu, 7 clipping, 206, 496, 683, 684 closure, 37, 109 coefficient binomial, 205 projection, 167, 472, 661, 668 cofactor, 440 compaction, 282, 291, 358, 452, 521 comparable, 93, 95, 166 complement algebraic, 87, 140 projection on, 648 orthogonal, 50 relative, 83, 708 complementarity, 199, 260 linear, 185 maximal, 248, 253, 276 problem, 185 strict, 261 complementary affine dimension, 485 dimension, 78 eigenvalues, 648 halfspaces, 68 inertia, 420, 558 slackness, 259 subspace, 50 completion geometry, 452 problem, 348, 405, 412, 443 semidefinite, 698 composition EDM, 414 compressed sensing, 201, 202, 325, 326, 328, 335, 336 nonnegative, 203, 297, 303–305 problem, 203, 296, 297, 303, 304 Procrustes, 314 compressive sampling, see compressed concave, 195, 242, 521, 597 problem, 195, 246 condition biorthogonality, 167, 563, 582 convexity first order, 228, 229, 232, 234 second order, 231, 236 optimality first order, 157, 182, 183 semidefinite program, 260 unconstrained, 157, 222, 223, 670 cone, √87, 90, 718 EDM, 451 blade, 151 boundary, 96 EDM, 447, 462 membership, 153, 258 PSD, 104, 109, 121, 474 ray, 97 circular, 92, 115, 116 right, 116 convex, 43, 91, 93, 99 dual, 140, 141, 186, 188, 189, 423 algorithm, 186 construction, 142, 143 examples, 149 facet, 165, 189 formula, 158, 159, 173, 181 halfspace-description, 158 in subspace, 170, 175 linear inequality, 153 Lorentz, 149, 161 membership relation, 152, 153 pointed, 135, 147 properties, 147 unique, 141, 165
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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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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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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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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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A.3. PROPER STATEMENTS 549 For A di
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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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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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A.5. EIGEN DECOMPOSITION 561 When B
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A.5. EIGEN DECOMPOSITION 563 dim N(
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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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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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B.4. AUXILIARY V -MATRICES 589 18.
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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 ∗
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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.
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Page 689 and 690:
E.10. ALTERNATING PROJECTION 689 b
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Page 691 and 692:
E.10. ALTERNATING PROJECTION 691 a
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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
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Page 701 and 702:
E.10. ALTERNATING PROJECTION 701 E.
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Page 703 and 704:
E.10. ALTERNATING PROJECTION 703 E
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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
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Page 709 and 710:
709 is or ← → t → 0 + as in
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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-
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Page 743 and 744:
BIBLIOGRAPHY 743 [306] Michael W. T
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Page 745 and 746:
[333] Margaret H. Wright. The inter
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Page 747:
Index 0-norm, 203, 261, 294, 296, 2
-
Page 751 and 752:
INDEX 751 coordinates, 140, 170, 17
-
Page 753 and 754:
INDEX 753 affine dimension, 485 fea
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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
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Page 761 and 762:
INDEX 761 strictly, 515, 520 functi
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Page 763 and 764:
INDEX 763 vector, 45, 241, 248, 325
-
Page 765 and 766:
INDEX 765 convex envelope, see conv
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Page 767 and 768:
INDEX 767 cone, 418, 420, 507 dual,
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Page 769 and 770:
INDEX 769 trilateration, 21, 42, 36
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Page 772:
Convex Optimization & Euclidean Dis
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