v2010.10.26 - Convex Optimization

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624 APPENDIX A. LINEAR ALGEBRAAn important geometrical interpretation of SVD is given in Figure 155for m = n = 2 : The image of the unit sphere under any m × n matrixmultiplication is an ellipse. Considering the three factors of the SVDseparately, note that Q T is a pure rotation of the circle. Figure 155 showshow the axes q 1 and q 2 are first rotated by Q T to coincide with the coordinateaxes. Second, the circle is stretched by Σ in the directions of the coordinateaxes to form an ellipse. The third step rotates the ellipse by U into itsfinal position. Note how q 1 and q 2 are rotated to end up as u 1 and u 2 , theprincipal axes of the final ellipse. A direct calculation shows that Aq j = σ j u j .Thus q j is first rotated to coincide with the j th coordinate axis, stretched bya factor σ j , and then rotated to point in the direction of u j . All of thisis beautifully illustrated for 2 ×2 matrices by the Matlab code eigshow.m(see [334]).A direct consequence of the geometric interpretation is that the largestsingular value σ 1 measures the “magnitude” of A (its 2-norm):‖A‖ 2 = sup ‖Ax‖ 2 = σ 1 (1574)‖x‖ 2 =1This means that ‖A‖ 2 is the length of the longest principal semiaxis of theellipse.Expressions for U , Q , and Σ follow readily from (1573),AA T U = UΣΣ T and A T AQ = QΣ T Σ (1575)demonstrating that the columns of U are the eigenvectors of AA T and thecolumns of Q are the eigenvectors of A T A. −Muller, Magaia, & Herbst [271]A.6.4Pseudoinverse by SVDMatrix pseudoinverse (E) is nearly synonymous with singular valuedecomposition because of the elegant expression, given A = UΣQ T ∈ R m×nA † = QΣ †T U T ∈ R n×m (1576)that applies to all three flavors of SVD, where Σ † simply inverts nonzeroentries of matrix Σ .Given symmetric matrix A∈ S n and its diagonalization A = SΛS T(A.5.1), its pseudoinverse simply inverts all nonzero eigenvalues:A † = SΛ † S T ∈ S n (1577)
A.7. ZEROS 625A.6.5SVD of symmetric matricesFrom (1566) and (1562) for A = A T⎧√ (√⎨ λ(A2 ) i = λ A 2)= |λ(A) i | > 0, 1 ≤ i ≤ ρσ(A) i =i⎩0, ρ < i ≤ η(1578)A.6.5.0.1 Definition. Step function. (confer4.3.2.0.1)Define the signum-like quasilinear function ψ : R n → R n that takes value 1corresponding to a 0-valued entry in its argument:[ψ(a) limx i →a ix i|x i | = { 1, ai ≥ 0−1, a i < 0 , i=1... n ]∈ R n (1579)Eigenvalue signs of a symmetric matrix having diagonalizationA = SΛS T (1559) can be absorbed either into real U or real Q from thefull SVD; [350, p.34] (conferC.4.2.1)△orA = SΛS T = Sδ(ψ(δ(Λ))) |Λ|S T U ΣQ T ∈ S n (1580)A = SΛS T = S|Λ|δ(ψ(δ(Λ)))S T UΣ Q T ∈ S n (1581)where matrix of singular values Σ = |Λ| denotes entrywise absolute value ofdiagonal eigenvalue matrix Λ .A.7 ZerosA.7.1norm zeroFor any given norm, by definition,‖x‖ l= 0 ⇔ x = 0 (1582)Consequently, a generally nonconvex constraint in x like ‖Ax − b‖ = κbecomes convex when κ = 0.
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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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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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Page 587 and 588: 7.3. THIRD PREVALENT PROBLEM: 587Op
Page 589 and 590: 7.4. CONCLUSION 589filtering, multi
Page 591 and 592: Appendix ALinear algebraA.1 Main-di
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Page 597 and 598: A.2. SEMIDEFINITENESS: DOMAIN OF TE
Page 599 and 600: A.3. PROPER STATEMENTS 599(AB) T
Page 601 and 602: A.3. PROPER STATEMENTS 601A.3.1Semi
Page 603 and 604: A.3. PROPER STATEMENTS 603For A dia
Page 605 and 606: A.3. PROPER STATEMENTS 605Diagonali
Page 607 and 608: A.3. PROPER STATEMENTS 607For A,B
Page 609 and 610: A.3. PROPER STATEMENTS 609When B is
Page 611 and 612: A.4. SCHUR COMPLEMENT 611A.4 Schur
Page 613 and 614: A.4. SCHUR COMPLEMENT 613A.4.0.0.3
Page 615 and 616: A.4. SCHUR COMPLEMENT 615From Corol
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Page 623: A.6. SINGULAR VALUE DECOMPOSITION,
Page 627 and 628: A.7. ZEROS 627(Transpose.)Likewise,
Page 629 and 630: A.7. ZEROS 629For X,A∈ S M +[34,2
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Page 633 and 634: Appendix BSimple matricesMathematic
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Page 637 and 638: B.1. RANK-ONE MATRIX (DYAD) 637B.1.
Page 639 and 640: B.2. DOUBLET 639R([u v ])R(Π)= R([
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Page 643 and 644: B.4. AUXILIARY V -MATRICES 643is an
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Page 647 and 648: B.5. ORTHOGONAL MATRIX 647Given X
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Page 651 and 652: B.5. ORTHOGONAL MATRIX 651which is
Page 653 and 654: Appendix CSome analytical optimal r
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Page 661 and 662: C.3. ORTHOGONAL PROCRUSTES PROBLEM
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Page 669 and 670: 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
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
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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
Page 815 and 816:
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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