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760 INDEX postulate, 347 property, 347, 400 fifth, 347, 350, 351, 435 space, 347 minimal element, 95 set, 59, 71, 130, 131, 660 minimax problem, 146 minimization, 157, 182, 282, 621 cardinality, 201, 202, 296, 323, 333 Boolean, 261 nonnegative, 203, 294, 297, 303, 304 signed, 306 on hypercube, 76 rank, 277, 310, 323, 338, 340, 523, 531 trace, 367, 518 minimum cardinality, see minimization element, 95 global unique, 195, 621 value unique, 198 molecular conformation, 22, 376 monotonic, 218, 225, 279, 693, 696 Fejér, 693 function, 195, 204, 205, 225, 242 gradient, 227 Moore-Penrose conditions, 639 MRI, 325, 326 Muller, 567 multidimensional function, 195, 221, 236 affine, 209 scaling, 22, 424 ordinal, 430 multilateration, 280, 373 multipath, 373 multiplicity, 549, 562 nearest neighbors, 454 neighborhood graph, 453, 454 nesting, 402 Newton, 361, 609 nondegeneracy, 347 nonexpansive, 651, 682 nonnegative, 398, 719 nonsymmetric, 547 nonvertical, 224 norm, 199, 294, 719 k largest, 205 ball, see ball constraint, 308, 317, 569 Euclidean, 45, 719 Frobenius, 47 minimization, 223, 670 inequality, 263, 720 triangle, 199 Ky Fan, 596 least, 201, 263, 642 nuclear, 596 optimization, 200, 219 regularization, 318, 642 spectral, 64, 500, 513, 567, 578, 585, 586, 684, 720 zero, 569 normal, 68, 183, 674 cone, 182, 681, 702 membership, 182 equation, 642 facet, 165 inward, 66, 477 outward, 66 vector, 674 Notation, 705 novelty, 30 NP-hard, 523 nuclear magnetic resonance, 376 null certificate, 154 nullspace, 78, 587, 672 basis, 80 doublet, 583, 584 dyad, 578, 579 elementary matrix, 584, 585 form, 78 hyperplane, 67 intersection, 81 orthogonal complement, 712 pseudoinverse, 639, 640 objective convex, 245
INDEX 761 strictly, 515, 520 function, 77, 146, 157, 182, 184 linear, 211, 288, 367 multidimensional, 198 nonlinear, 211 polynomial, 315 quadratic, 219 real, 198 value, 259, 273 unique, 247 offset, 382 on, 712 onto, 712 operator adjoint, 45, 148, 153, 484, 537 invertible, 48, 49 linear, 43, 48, 53, 196, 209, 211, 355, 389, 391, 433, 467, 537, 672 projector, 647, 652, 654, 675 nonlinear, 196 convex, 211 nullspace, 383, 387, 389, 391, 672 permutation, 204, 295, 507, 511, 532, 711 unitary, 48 optimality condition first order, 157, 182, 183 semidefinite program, 260 unconstrained, 157, 222, 223, 670 conic constraint, 184 equality constraint, 157 global, 245 perturbation, 273 optimization, 245 combinatorial, 63, 76, 204, 311, 315, 318, 409, 536 convex, 7, 198, 246, 365 art, 8, 267, 282, 536 fundamental, 245 multicriteria, 198 multiobjective, 198, 294, 333 norm, 200, 219 tractable, 246 vector, 198 order natural, 45, 537, 716 nonincreasing, 509, 564, 565, 711 of projection, 496 partial, 57, 91, 93–95, 105, 160, 165, 166, 552, 719 generalized inequality, 93 total, 93 origin, 34 cone, 64, 90, 93, 97, 99, 127 projection of, 201, 202, 643, 659 projection on, 679, 684 translation to, 394 Orion nebula, 20 orthant, 35, 137, 169, 182 nonnegative, 153, 483 orthogonal, 45 complement, 50 equivalence, 594 expansion, 52, 140, 169, 651, 653 invariance, 48, 433 orthonormal, 45 decomposition, 650 matrix, see matrix orthonormality condition, 169 overdetermined, 705 parallel, 36 pattern recognition, 22 Penrose conditions, 639 pentahedron, 440 pentatope, 440 perpendicular, 45, 649, 670 perturbation, 268, 270 optimality, 273 phantom, 326 plane, 66 segment, 97 point boundary, 85 hypercube slice, 336 fixed, 279, 316, 334, 672, 690–693 inflection, 231 polychoron, 87, 440 polyhedron, 39, 55, 76, 132, 406 bounded, 59, 203
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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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2.3. 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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2.10. CONIC INDEPENDENCE (C.I.) 127
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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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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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3.3. 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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4.5. 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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4.7. 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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5.7. 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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5.9. 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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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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6.7. 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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7.3. 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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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.3. PROPER STATEMENTS 545 (AB) T
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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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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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B.1. RANK-ONE MATRIX (DYAD) 579 R(v
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B.1. RANK-ONE MATRIX (DYAD) 581 ran
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B.2. DOUBLET 583 R([u v ]) R(Π)= R
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B.3. ELEMENTARY MATRIX 585 If λ
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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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Appendix C Some analytical optimal
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C.2. TRACE, SINGULAR AND EIGEN VALU
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C.3. ORTHOGONAL PROCRUSTES PROBLEM
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Appendix D Matrix calculus From too
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D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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D.2. TABLES OF GRADIENTS AND DERIVA
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Appendix E Projection For any A∈
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641 U T = U † for orthonormal (in
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E.1. IDEMPOTENT MATRICES 643 where
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E.1. IDEMPOTENT MATRICES 645 order,
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E.1. IDEMPOTENT MATRICES 647 When t
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.5. PROJECTION EXAMPLES 655 E.4.0.
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E.5. PROJECTION EXAMPLES 657 a ∗
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E.5. PROJECTION EXAMPLES 659 E.5.0.
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E.6. VECTORIZATION INTERPRETATION,
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E.7. ON VECTORIZED MATRICES OF HIGH
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E.7. ON VECTORIZED MATRICES OF HIGH
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E.8. RANGE/ROWSPACE INTERPRETATION
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E.9. PROJECTION ON CONVEX SET 675 A
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E.9. PROJECTION ON CONVEX SET 677 W
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E.9. PROJECTION ON CONVEX SET 679 P
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E.9. PROJECTION ON CONVEX SET 681 E
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E.9. PROJECTION ON CONVEX SET 683 T
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E.9. PROJECTION ON CONVEX SET 685
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E.10. ALTERNATING PROJECTION 687 E.
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E.10. ALTERNATING PROJECTION 689 b
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E.10. ALTERNATING PROJECTION 691 a
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E.10. ALTERNATING PROJECTION 693 (a
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E.10. ALTERNATING PROJECTION 695 wh
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E.10. ALTERNATING PROJECTION 699 10
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E.10. ALTERNATING PROJECTION 703 E
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Appendix F Notation and a few defin
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707 a.i. c.i. l.i. w.r.t affinely i
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Page 717 and 718: 717 O O sort-index matrix order of
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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: INDEX 759 normal, 47, 548, 563 norm
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,
Page 769 and 770: INDEX 769 trilateration, 21, 42, 36
Page 772: Convex Optimization & Euclidean Dis
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v2009.01.01 - Convex Optimization

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