v2010.10.26 - Convex Optimization

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438 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXTo maintain relative quadrature between points (Figure 128a) and to preventreflection, it is sufficient that all interpoint distances be specified and thatadjacencies Y (:, 1:2) , Y (:, 2:3) , and Y (:, 3:4) be proportional to 2×2rotation matrices; any clockwise rotation would ascribe a reflection matrixcharacteristic. Counterclockwise rotation is thereby enforced by constrainingequality among diagonal and antidiagonal entries as prescribed by (981);[ ] 0 1Y (:,1:3) = Y (:, 2:4) (985)−1 0Quadrature quantization of rotation can be regarded as a constrainton tilt of the smallest Cartesian square containing the diamond as inFigure 128c. Our scheme to quantize rotation requires that all squarevertices be described by vectors whose entries are nonnegative when thesquare is translated anywhere interior to the nonnegative orthant. Wecapture the four square vertices as columns of a product Y C where⎡ ⎤1 0 0 1C = ⎢ 1 1 0 0⎥⎣ 0 1 1 0 ⎦ (986)0 0 1 1Then, assuming a unit-square shroud, the affine constraint[ ] 1/2Y C + 1 T ≥ 0 (987)1/2quantizes rotation, as desired.5.5.2.1 Inner-product form invarianceLikewise, D(Θ) (958) is rotation/reflection invariant;so (979) and (980) similarly apply.5.5.3 Invariance conclusionD(Q p Θ) = D(QΘ) = D(Θ) (988)In the making of an EDM, absolute rotation, reflection, and translationinformation is lost. Given an EDM, reconstruction of point position (5.12,the list X) can be guaranteed correct only in affine dimension r and relativeposition. Given a noiseless complete EDM, this isometric reconstruction isunique in so far as every realization of a corresponding list X is congruent:
5.6. INJECTIVITY OF D & UNIQUE RECONSTRUCTION 4395.6 Injectivity of D & unique reconstructionInjectivity implies uniqueness of isometric reconstruction (5.4.2.3.7); hence,we endeavor to demonstrate it.EDM operators list-form D(X) (891), Gram-form D(G) (903), andinner-product form D(Θ) (958) are many-to-one surjections (5.5) onto thesame range; the EDM cone (6): (confer (904) (995))EDM N = { D(X) : R N−1×N → S N h | X ∈ R N−1×N}= { }D(G) : S N → S N h | G ∈ S N + − S N⊥c= { (989)D(Θ) : R N−1×N−1 → S N h | Θ ∈ R N−1×N−1}where (5.5.1.1)S N⊥c = {u1 T + 1u T | u∈ R N } ⊆ S N (2000)5.6.1 Gram-form bijectivityBecause linear Gram-form EDM operatorD(G) = δ(G)1 T + 1δ(G) T − 2G (903)has no nullspace [85,A.1] on the geometric center subspace 5.27 (E.7.2.0.2)S N c {G∈ S N | G1 = 0} (1998)= {G∈ S N | N(G) ⊇ 1} = {G∈ S N | R(G) ⊆ N(1 T )}= {V Y V | Y ∈ S N } ⊂ S N (1999)≡ {V N AVN T | A ∈ SN−1 }(990)then D(G) on that subspace is injective.5.27 The equivalence ≡ in (990) follows from the fact: Given B = V Y V = V N AVN T ∈ SN cwith only matrix A∈ S N−1 unknown, then V † †TNBV N = A or V † N Y V †TN = A.
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DATTORROCONVEXOPTIMIZATION&EUCLIDEA
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Convex Optimization&Euclidean Dista
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for Jennie Columba♦Antonio♦♦&
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PreludeThe constant demands of my d
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Convex Optimization&Euclidean Dista
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures1 Overview 211 Orion
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LIST OF FIGURES 1562 Shrouded polyh
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LIST OF FIGURES 17130 Elliptope E 3
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Chapter 1OverviewConvex Optimizatio
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ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
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25Figure 4: This coarsely discretiz
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(biorthogonal expansion) is examine
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29cardinality Boolean solution to a
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31Figure 8: Robotic vehicles in con
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an elaborate exposition offering in
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Chapter 2Convex geometryConvexity h
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2.1. CONVEX SET 372.1.2 linear inde
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2.1. CONVEX SET 392.1.6 empty set v
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2.1. CONVEX SET 41(a)R(b)R 2(c)R 3F
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2.1. CONVEX SET 43where Q∈ R 3×3
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2.1. CONVEX SET 45Now let’s move
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2.1. CONVEX SET 47By additive inver
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2.1. CONVEX SET 49R nR mR(A T )x px
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 61Figure 20: Convex hull
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2.3. HULLS 63Aaffine hull (drawn tr
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2.3. HULLS 65subset of the affine h
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2.3. HULLS 672.3.2.0.2 Example. Nuc
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2.3. HULLS 692.3.2.0.3 Exercise. Co
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2.3. HULLS 71Figure 24: A simplicia
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2.4. HALFSPACE, HYPERPLANE 73H + =
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2.4. HALFSPACE, HYPERPLANE 7511−1
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2.4. HALFSPACE, HYPERPLANE 772.4.2.
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2.4. HALFSPACE, HYPERPLANE 81tradit
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2.5. SUBSPACE REPRESENTATIONS 852.5
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2.5. SUBSPACE REPRESENTATIONS 87(Ex
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2.5. SUBSPACE REPRESENTATIONS 89(a)
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2.5. SUBSPACE REPRESENTATIONS 91are
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2.6. EXTREME, EXPOSED 93A one-dimen
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2.6. EXTREME, EXPOSED 952.6.1.1 Den
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2.7. CONES 97X(a)00(b)XFigure 33: (
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2.7. CONES 99XXFigure 37: Truncated
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2.7. CONES 101Figure 39: Not a cone
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2.7. CONES 103cone that is a halfli
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2.7. CONES 105A pointed closed conv
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2.8. CONE BOUNDARY 107That means th
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2.8. CONE BOUNDARY 1092.8.1.1 extre
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2.8. CONE BOUNDARY 1112.8.2 Exposed
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 141
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2.12. CONVEX POLYHEDRA 147all dimen
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2.12. CONVEX POLYHEDRA 149convex po
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2.12. CONVEX POLYHEDRA 1512.12.2.2
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3Geometry of convex functio
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3.1. CONVEX FUNCTION 221f 1 (x)f 2
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3.1. CONVEX FUNCTION 223Rf(b)f(X
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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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3.3. INVERTED FUNCTIONS AND ROOTS 2
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3.4. AFFINE FUNCTION 237rather]x >
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3.4. AFFINE FUNCTION 239f(z)Az 2z 1
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3.5. EPIGRAPH, SUBLEVEL SET 241{a T
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3.5. EPIGRAPH, SUBLEVEL SET 243Subl
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3.5. EPIGRAPH, SUBLEVEL SET 245wher
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3.5. EPIGRAPH, SUBLEVEL SET 247part
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3.5. EPIGRAPH, SUBLEVEL SET 249that
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3.6. GRADIENT 251respect to its vec
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3.6. GRADIENT 253Invertibility is g
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3.6. GRADIENT 2553.6.1.0.2 Theorem.
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3.6. GRADIENT 257f(Y )[ ∇f(X)−1
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3.6. GRADIENT 259αβα ≥ β ≥
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3.6. GRADIENT 2613.6.4 second-order
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3.7. CONVEX MATRIX-VALUED FUNCTION
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3.8. QUASICONVEX 269exponential alw
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3.9. SALIENT PROPERTIES 2713.8.0.0.
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Chapter 4Semidefinite programmingPr
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4.1. CONIC PROBLEM 275(confer p.162
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4.1. CONIC PROBLEM 277PCsemidefinit
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4.1. CONIC PROBLEM 279is the affine
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4.1. CONIC PROBLEM 281faces of S 3
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4.1. CONIC PROBLEM 2834.1.2.3 Previ
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4.2. FRAMEWORK 285Semidefinite Fark
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4.2. FRAMEWORK 287On the other hand
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4.2. FRAMEWORK 2894.2.2.1 Dual prob
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4.2. FRAMEWORK 291For symmetric pos
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4.2. FRAMEWORK 293has norm ‖x ⋆
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4.2. FRAMEWORK 295minimize 1 TˆxX
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4.2. FRAMEWORK 297asminimize ‖ỹ
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4.3. RANK REDUCTION 2994.3 Rank red
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4.3. RANK REDUCTION 301A rank-reduc
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4.3. RANK REDUCTION 303(t ⋆ i)
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4.3. RANK REDUCTION 3054.3.3.0.1 Ex
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4.3. RANK REDUCTION 3074.3.3.0.2 Ex
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 333mi
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4.5. CONSTRAINING CARDINALITY 3350R
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4.5. CONSTRAINING CARDINALITY 337it
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4.5. CONSTRAINING CARDINALITY 339m/
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4.5. CONSTRAINING CARDINALITY 341we
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4.5. CONSTRAINING CARDINALITY 343fl
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4.5. CONSTRAINING CARDINALITY 345We
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4.5. CONSTRAINING CARDINALITY 3474.
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4.5. CONSTRAINING CARDINALITY 349R
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4.5. CONSTRAINING CARDINALITY 351pe
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.7. CONSTRAINING RANK OF INDEFINIT
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Page 391 and 392: 4.8. CONVEX ITERATION RANK-1 391whi
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Page 395 and 396: Chapter 5Euclidean Distance MatrixT
Page 397 and 398: 5.2. FIRST METRIC PROPERTIES 397to
Page 399 and 400: 5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 401 and 402: 5.3. ∃ FIFTH EUCLIDEAN METRIC PRO
Page 403 and 404: 5.4. EDM DEFINITION 403The collecti
Page 405 and 406: 5.4. EDM DEFINITION 4055.4.2 Gram-f
Page 407 and 408: 5.4. EDM DEFINITION 407We provide a
Page 409 and 410: 5.4. EDM DEFINITION 4095.4.2.3.1 Ex
Page 411 and 412: 5.4. EDM DEFINITION 411is the fact:
Page 413 and 414: 5.4. EDM DEFINITION 413Figure 120:
Page 415 and 416: 5.4. EDM DEFINITION 415is found fro
Page 417 and 418: 5.4. EDM DEFINITION 417one less dim
Page 419 and 420: 5.4. EDM DEFINITION 419equality con
Page 421 and 422: 5.4. EDM DEFINITION 421How much dis
Page 423 and 424: 5.4. EDM DEFINITION 423105ˇx 4ˇx
Page 425 and 426: 5.4. EDM DEFINITION 425now implicit
Page 427 and 428: 5.4. EDM DEFINITION 427by translate
Page 429 and 430: 5.4. EDM DEFINITION 429Crippen & Ha
Page 431 and 432: 5.4. EDM DEFINITION 431where ([√t
Page 433 and 434: 5.4. EDM DEFINITION 433because (A.3
Page 435 and 436: 5.5. INVARIANCE 4355.5.1.0.1 Exampl
Page 437: 5.5. INVARIANCE 437x 2 x 2x 3 x 1 x
Page 441 and 442: 5.6. INJECTIVITY OF D & UNIQUE RECO
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Page 445 and 446: 5.7. EMBEDDING IN AFFINE HULL 4455.
Page 447 and 448: 5.7. EMBEDDING IN AFFINE HULL 447Fo
Page 449 and 450: 5.7. EMBEDDING IN AFFINE HULL 4495.
Page 451 and 452: 5.8. EUCLIDEAN METRIC VERSUS MATRIX
Page 459 and 460: 5.9. BRIDGE: CONVEX POLYHEDRA TO ED
Page 465 and 466: 5.10. EDM-ENTRY COMPOSITION 465of a
Page 467 and 468: 5.10. EDM-ENTRY COMPOSITION 467Then
Page 469 and 470: 5.11. EDM INDEFINITENESS 4695.11.1
Page 471 and 472: 5.11. EDM INDEFINITENESS 471we have
Page 473 and 474: 5.11. EDM INDEFINITENESS 473So beca
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Page 477 and 478: 5.12. LIST RECONSTRUCTION 477where
Page 479 and 480: 5.12. LIST RECONSTRUCTION 479(a)(c)
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6Cone of distance matricesF
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6.1. DEFINING EDM CONE 4996.1 Defin
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6.2. POLYHEDRAL BOUNDS 501This cone
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6.4. EDM DEFINITION IN 11 T 503That
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6.4. EDM DEFINITION IN 11 T 505N(1
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6.4. EDM DEFINITION IN 11 T 507Then
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6.4. EDM DEFINITION IN 11 T 5096.4.
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6.5. CORRESPONDENCE TO PSD CONE S N
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6.6. VECTORIZATION & PROJECTION INT
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6.7. A GEOMETRY OF COMPLETION 523(b
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6.7. A GEOMETRY OF COMPLETION 525[3
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6.7. A GEOMETRY OF COMPLETION 527wh
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6.8. DUAL EDM CONE 529to the geomet
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6.8. DUAL EDM CONE 531Proof. First,
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6.8. DUAL EDM CONE 533EDM 2 = S 2 h
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6.8. DUAL EDM CONE 535therefore the
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6.8. DUAL EDM CONE 537Elegance of t
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6.8. DUAL EDM CONE 5396.8.1.5 Affin
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6.8. DUAL EDM CONE 5416.8.1.7 Schoe
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6.8. DUAL EDM CONE 5430dvec rel ∂
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6.10. POSTSCRIPT 5456.10 Postscript
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Chapter 7Proximity problemsIn the
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549project on the subspace, then pr
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551HS N h0EDM NK = S N h ∩ R N×N
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5537.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 555fi
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7.1. FIRST PREVALENT PROBLEM: 5577.
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7.1. FIRST PREVALENT PROBLEM: 559di
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7.1. FIRST PREVALENT PROBLEM: 5617.
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7.1. FIRST PREVALENT PROBLEM: 563wh
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7.1. FIRST PREVALENT PROBLEM: 565Th
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7.2. SECOND PREVALENT PROBLEM: 567O
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7.2. SECOND PREVALENT PROBLEM: 569S
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7.2. SECOND PREVALENT PROBLEM: 571r
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7.2. SECOND PREVALENT PROBLEM: 573w
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7.2. SECOND PREVALENT PROBLEM: 5757
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7.2. SECOND PREVALENT PROBLEM: 577a
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7.3. THIRD PREVALENT PROBLEM: 579is
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7.3. THIRD PREVALENT PROBLEM: 581We
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7.3. THIRD PREVALENT PROBLEM: 583su
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7.3. THIRD PREVALENT PROBLEM: 585Gi
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7.3. THIRD PREVALENT PROBLEM: 587Op
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7.4. CONCLUSION 589filtering, multi
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Appendix ALinear algebraA.1 Main-di
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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 599(AB) T
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A.3. PROPER STATEMENTS 601A.3.1Semi
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A.3. PROPER STATEMENTS 603For A dia
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A.3. PROPER STATEMENTS 605Diagonali
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A.3. PROPER STATEMENTS 607For A,B
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A.3. PROPER STATEMENTS 609When B is
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A.4. SCHUR COMPLEMENT 611A.4 Schur
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A.4. SCHUR COMPLEMENT 613A.4.0.0.3
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A.4. SCHUR COMPLEMENT 615From Corol
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A.5. EIGENVALUE DECOMPOSITION 617wh
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A.5. EIGENVALUE DECOMPOSITION 619A.
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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 625A.6.5SVD of symmetric
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A.7. ZEROS 627(Transpose.)Likewise,
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A.7. ZEROS 629For X,A∈ S M +[34,2
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A.7. ZEROS 631A.7.5.0.1 Proposition
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Appendix BSimple matricesMathematic
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B.1. RANK-ONE MATRIX (DYAD) 635R(v)
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B.1. RANK-ONE MATRIX (DYAD) 637B.1.
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B.2. DOUBLET 639R([u v ])R(Π)= R([
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B.3. ELEMENTARY MATRIX 641has N −
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B.4. AUXILIARY V -MATRICES 643is an
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B.4. AUXILIARY V -MATRICES 64514. [
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B.5. ORTHOGONAL MATRIX 647Given X
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B.5. ORTHOGONAL MATRIX 649Figure 15
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B.5. ORTHOGONAL MATRIX 651which is
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Appendix CSome analytical optimal r
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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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Appendix DMatrix calculusFrom too m
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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 EProjectionFor any A∈ R
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701U T = U † for orthonormal (inc
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E.1. IDEMPOTENT MATRICES 703where A
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E.1. IDEMPOTENT MATRICES 705order,
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E.1. IDEMPOTENT MATRICES 707are lin
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.4. ALGEBRA OF PROJECTION ON AFFIN
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E.5. PROJECTION EXAMPLES 717a ∗ 2
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E.5. PROJECTION EXAMPLES 719where Y
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E.5. PROJECTION EXAMPLES 721(B.4.2)
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E.6. VECTORIZATION INTERPRETATION,
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E.7. PROJECTION ON MATRIX SUBSPACES
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E.7. PROJECTION ON MATRIX SUBSPACES
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E.8. RANGE/ROWSPACE INTERPRETATION
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E.9. PROJECTION ON CONVEX SET 735As
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E.9. PROJECTION ON CONVEX SET 737Wi
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E.9. PROJECTION ON CONVEX SET 739R(
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E.9. PROJECTION ON CONVEX SET 741E.
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E.9. PROJECTION ON CONVEX SET 743E.
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E.9. PROJECTION ON CONVEX SET 745Un
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E.9. PROJECTION ON CONVEX SET 747ac
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E.10. ALTERNATING PROJECTION 749bC
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E.10. ALTERNATING PROJECTION 7510
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E.10. ALTERNATING PROJECTION 753E.1
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E.10. ALTERNATING PROJECTION 755y 2
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E.10. ALTERNATING PROJECTION 757Def
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E.10. ALTERNATING PROJECTION 759Dis
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E.10. ALTERNATING PROJECTION 761mat
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E.10. ALTERNATING PROJECTION 763K
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Appendix FNotation and a few defini
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771A ij or A(i, j) , ij th entry of
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773⊞orthogonal vector sum of sets
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775x +vector x whose negative entri
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777X point list ((76) having cardin
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779SDPSVDSNRdBEDMS n 1S n hS n⊥hS
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781vectorentrycubixquartixfeasible
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783Oorder of magnitude information
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785cofmatrix of cofactors correspon
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Bibliography[1] Edwin A. Abbott. Fl
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BIBLIOGRAPHY 789[23] Dror Baron, Mi
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BIBLIOGRAPHY 791[49] Leonard M. Blu
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BIBLIOGRAPHY 793[74] Yves Chabrilla
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BIBLIOGRAPHY 795[102] Etienne de Kl
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BIBLIOGRAPHY 797[129] Carl Eckart a
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BIBLIOGRAPHY 799[154] James Gleik.
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BIBLIOGRAPHY 801[182] Johan Håstad
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BIBLIOGRAPHY 803[212] Viren Jain an
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BIBLIOGRAPHY 805[237] Monique Laure
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BIBLIOGRAPHY 807[265] Sunderarajan
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BIBLIOGRAPHY 809Notes in Computer S
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BIBLIOGRAPHY 811[319] Anthony Man-C
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BIBLIOGRAPHY 813[346] Pham Dinh Tao
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BIBLIOGRAPHY 815[375] Bernard Widro
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Index∅, see empty set0-norm, 229,
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INDEX 819bees, 30, 413Bellman, 276b
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INDEX 821circular, 130construction,
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INDEX 823measure, 295convex, 21, 36
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INDEX 825duality, 160, 410gap, 161,
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INDEX 827positive semidefinite, 284
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INDEX 829point, 39, 62positive semi
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INDEX 831isomorphic, 52, 56, 59, 77
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INDEX 833nonsymmetric, 703range, 70
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INDEX 835Muller, 624multidimensiona
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INDEX 837projection on, 739, 745tra
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INDEX 839pseudoinverse, 701trace, 5
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INDEX 841coordinate system, 156line
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INDEX 843nonconvex, 40, 97-101nulls
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INDEX 845faces, 106intersection, 10
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INDEX 847projection, 517symmetric,
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849
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Convex Optimization & Euclidean Dis
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