v2010.10.26 - Convex Optimization

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10 CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY3.7 Convex matrix-valued function . . . . . . . . . . . . . . . . . . 2623.8 Quasiconvex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2693.9 Salient properties . . . . . . . . . . . . . . . . . . . . . . . . . 2714 Semidefinite programming 2734.1 Conic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2744.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2844.3 Rank reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2994.4 Rank-constrained semidefinite program . . . . . . . . . . . . . 3084.5 Constraining cardinality . . . . . . . . . . . . . . . . . . . . . 3334.6 Cardinality and rank constraint examples . . . . . . . . . . . . 3534.7 Constraining rank of indefinite matrices . . . . . . . . . . . . . 3844.8 Convex Iteration rank-1 . . . . . . . . . . . . . . . . . . . . . 3905 Euclidean Distance Matrix 3955.1 EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3965.2 First metric properties . . . . . . . . . . . . . . . . . . . . . . 3975.3 ∃ fifth Euclidean metric property . . . . . . . . . . . . . . . . 3975.4 EDM definition . . . . . . . . . . . . . . . . . . . . . . . . . . 4025.5 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4345.6 Injectivity of D & unique reconstruction . . . . . . . . . . . . 4395.7 Embedding in affine hull . . . . . . . . . . . . . . . . . . . . . 4455.8 Euclidean metric versus matrix criteria . . . . . . . . . . . . . 4505.9 Bridge: Convex polyhedra to EDMs . . . . . . . . . . . . . . . 4585.10 EDM-entry composition . . . . . . . . . . . . . . . . . . . . . 4655.11 EDM indefiniteness . . . . . . . . . . . . . . . . . . . . . . . . 4685.12 List reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 4765.13 Reconstruction examples . . . . . . . . . . . . . . . . . . . . . 4805.14 Fifth property of Euclidean metric . . . . . . . . . . . . . . . 4866 Cone of distance matrices 4976.1 Defining EDM cone . . . . . . . . . . . . . . . . . . . . . . . . 4996.2 √Polyhedral bounds . . . . . . . . . . . . . . . . . . . . . . . . 5016.3 EDM cone is not convex . . . . . . . . . . . . . . . . . . . . 5026.4 EDM definition in 11 T . . . . . . . . . . . . . . . . . . . . . . 5036.5 Correspondence to PSD cone S N−1+ . . . . . . . . . . . . . . . 5116.6 Vectorization & projection interpretation . . . . . . . . . . . . 5176.7 A geometry of completion . . . . . . . . . . . . . . . . . . . . 522
CONVEX OPTIMIZATION & EUCLIDEAN DISTANCE GEOMETRY 116.8 Dual EDM cone . . . . . . . . . . . . . . . . . . . . . . . . . . 5296.9 Theorem of the alternative . . . . . . . . . . . . . . . . . . . . 5446.10 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5457 Proximity problems 5477.1 First prevalent problem: . . . . . . . . . . . . . . . . . . . . . 5557.2 Second prevalent problem: . . . . . . . . . . . . . . . . . . . . 5667.3 Third prevalent problem: . . . . . . . . . . . . . . . . . . . . . 5787.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588A Linear algebra 591A.1 Main-diagonal δ operator, λ , trace, vec . . . . . . . . . . . . 591A.2 Semidefiniteness: domain of test . . . . . . . . . . . . . . . . . 596A.3 Proper statements . . . . . . . . . . . . . . . . . . . . . . . . . 599A.4 Schur complement . . . . . . . . . . . . . . . . . . . . . . . . 611A.5 Eigenvalue decomposition . . . . . . . . . . . . . . . . . . . . 616A.6 Singular value decomposition, SVD . . . . . . . . . . . . . . . 620A.7 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625B Simple matrices 633B.1 Rank-one matrix (dyad) . . . . . . . . . . . . . . . . . . . . . 634B.2 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639B.3 Elementary matrix . . . . . . . . . . . . . . . . . . . . . . . . 640B.4 Auxiliary V -matrices . . . . . . . . . . . . . . . . . . . . . . . 642B.5 Orthogonal matrix . . . . . . . . . . . . . . . . . . . . . . . . 647C Some analytical optimal results 653C.1 Properties of infima . . . . . . . . . . . . . . . . . . . . . . . . 653C.2 Trace, singular and eigen values . . . . . . . . . . . . . . . . . 654C.3 Orthogonal Procrustes problem . . . . . . . . . . . . . . . . . 661C.4 Two-sided orthogonal Procrustes . . . . . . . . . . . . . . . . 663C.5 Nonconvex quadratics . . . . . . . . . . . . . . . . . . . . . . 668D Matrix calculus 669D.1 Directional derivative, Taylor series . . . . . . . . . . . . . . . 669D.2 Tables of gradients and derivatives . . . . . . . . . . . . . . . 690
Page 1 and 2: DATTORROCONVEXOPTIMIZATION&EUCLIDEA
Page 3 and 4: Convex Optimization&Euclidean Dista
Page 5 and 6: for Jennie Columba♦Antonio♦♦&
Page 7 and 8: PreludeThe constant demands of my d
Page 9: Convex Optimization&Euclidean Dista
Page 13 and 14: List of Figures1 Overview 211 Orion
Page 15 and 16: LIST OF FIGURES 1562 Shrouded polyh
Page 17: LIST OF FIGURES 17130 Elliptope E 3
Page 21 and 22: Chapter 1OverviewConvex Optimizatio
Page 23 and 24: ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
Page 25 and 26: 25Figure 4: This coarsely discretiz
Page 27 and 28: (biorthogonal expansion) is examine
Page 29 and 30: 29cardinality Boolean solution to a
Page 31 and 32: 31Figure 8: Robotic vehicles in con
Page 33 and 34: an elaborate exposition offering in
Page 35 and 36: Chapter 2Convex geometryConvexity h
Page 37 and 38: 2.1. CONVEX SET 372.1.2 linear inde
Page 39 and 40: 2.1. CONVEX SET 392.1.6 empty set v
Page 41 and 42: 2.1. CONVEX SET 41(a)R(b)R 2(c)R 3F
Page 43 and 44: 2.1. CONVEX SET 43where Q∈ R 3×3
Page 45 and 46: 2.1. CONVEX SET 45Now let’s move
Page 47 and 48: 2.1. CONVEX SET 47By additive inver
Page 49 and 50: 2.1. CONVEX SET 49R nR mR(A T )x px
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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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4.8. CONVEX ITERATION RANK-1 391whi
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4.8. CONVEX ITERATION RANK-1 393the
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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 397to
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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 403The collecti
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5.4. EDM DEFINITION 4055.4.2 Gram-f
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5.4. EDM DEFINITION 407We provide a
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5.4. EDM DEFINITION 4095.4.2.3.1 Ex
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5.4. EDM DEFINITION 411is the fact:
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5.4. EDM DEFINITION 413Figure 120:
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5.4. EDM DEFINITION 415is found fro
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5.4. EDM DEFINITION 417one less dim
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5.4. EDM DEFINITION 419equality con
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5.4. EDM DEFINITION 421How much dis
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5.4. EDM DEFINITION 423105ˇx 4ˇx
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5.4. EDM DEFINITION 425now implicit
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5.4. EDM DEFINITION 427by translate
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5.4. EDM DEFINITION 429Crippen & Ha
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5.4. EDM DEFINITION 431where ([√t
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5.4. EDM DEFINITION 433because (A.3
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5.5. INVARIANCE 4355.5.1.0.1 Exampl
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5.5. INVARIANCE 437x 2 x 2x 3 x 1 x
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 4455.
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5.7. EMBEDDING IN AFFINE HULL 447Fo
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5.7. EMBEDDING IN AFFINE HULL 4495.
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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 465of a
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5.10. EDM-ENTRY COMPOSITION 467Then
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5.11. EDM INDEFINITENESS 4695.11.1
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5.11. EDM INDEFINITENESS 471we have
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5.11. EDM INDEFINITENESS 473So beca
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5.11. EDM INDEFINITENESS 475holds o
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5.12. LIST RECONSTRUCTION 477where
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5.12. LIST RECONSTRUCTION 479(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 481Wi
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5.13. RECONSTRUCTION EXAMPLES 483d
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5.13. RECONSTRUCTION EXAMPLES 485Th
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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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Page 697 and 698:
Page 699 and 700:
Appendix EProjectionFor any A∈ R
Page 701 and 702:
701U T = U † for orthonormal (inc
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E.1. IDEMPOTENT MATRICES 703where A
Page 705 and 706:
E.1. IDEMPOTENT MATRICES 705order,
Page 707 and 708:
E.1. IDEMPOTENT MATRICES 707are lin
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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Page 713 and 714:
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E.4. ALGEBRA OF PROJECTION ON AFFIN
Page 717 and 718:
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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Page 727 and 728:
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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
Page 735 and 736:
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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Page 767 and 768:
Page 769 and 770:
Appendix FNotation and a few defini
Page 771 and 772:
771A ij or A(i, j) , ij th entry of
Page 773 and 774:
773⊞orthogonal vector sum of sets
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775x +vector x whose negative entri
Page 777 and 778:
777X point list ((76) having cardin
Page 779 and 780:
779SDPSVDSNRdBEDMS n 1S n hS n⊥hS
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781vectorentrycubixquartixfeasible
Page 783 and 784:
783Oorder of magnitude information
Page 785 and 786:
785cofmatrix of cofactors correspon
Page 787 and 788:
Bibliography[1] Edwin A. Abbott. Fl
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BIBLIOGRAPHY 789[23] Dror Baron, Mi
Page 791 and 792:
BIBLIOGRAPHY 791[49] Leonard M. Blu
Page 793 and 794:
BIBLIOGRAPHY 793[74] Yves Chabrilla
Page 795 and 796:
BIBLIOGRAPHY 795[102] Etienne de Kl
Page 797 and 798:
BIBLIOGRAPHY 797[129] Carl Eckart a
Page 799 and 800:
BIBLIOGRAPHY 799[154] James Gleik.
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BIBLIOGRAPHY 801[182] Johan Håstad
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BIBLIOGRAPHY 803[212] Viren Jain an
Page 805 and 806:
BIBLIOGRAPHY 805[237] Monique Laure
Page 807 and 808:
BIBLIOGRAPHY 807[265] Sunderarajan
Page 809 and 810:
BIBLIOGRAPHY 809Notes in Computer S
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BIBLIOGRAPHY 811[319] Anthony Man-C
Page 813 and 814:
BIBLIOGRAPHY 813[346] Pham Dinh Tao
Page 815 and 816:
BIBLIOGRAPHY 815[375] Bernard Widro
Page 817 and 818:
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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