v2007.11.26 - Convex Optimization

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554 APPENDIX C. SOME ANALYTICAL OPTIMAL RESULTSthe largest eigenvalue, and v N a normalized eigenvector correspondingto the smallest eigenvalue,v N = arg inf x T Ax (1499)‖x‖=1v 1 = arg sup x T Ax (1500)‖x‖=1For A∈ S N having eigenvalues λ(A)∈ R N , consider the unconstrainednonconvex optimization that is a projection on the rank-1 subset(2.9.2.1) of the boundary of positive semidefinite cone S N + : Definingλ 1 ∆ = maxi{λ(A) i } and corresponding eigenvector v 1minimizex‖xx T − A‖ 2 F = minimize tr(xx T (x T x) − 2Axx T + A T A)x{‖λ(A)‖ 2 , λ 1 ≤ 0=(1501)‖λ(A)‖ 2 − λ 2 1 , λ 1 > 0arg minimizex‖xx T − A‖ 2 F ={0 , λ1 ≤ 0v 1√λ1 , λ 1 > 0(1502)Proof. This is simply the Eckart & Young solution from7.1.2:x ⋆ x ⋆T ={0 , λ1 ≤ 0λ 1 v 1 v T 1 , λ 1 > 0(1503)minimizexGiven nonincreasingly ordered diagonalization A = QΛQ TΛ = δ(λ(A)) (A.5), then (1501) has minimum valuewhere⎧‖QΛQ T ‖ 2 F = ‖δ(Λ)‖2 , λ 1 ≤ 0⎪⎨⎛⎡‖xx T −A‖ 2 F =λ 1 Q ⎜⎢0⎝⎣. ..⎪⎩ ∥0⎤ ⎞ ∥ ∥∥∥∥∥∥2⎡⎥ ⎟⎦− Λ⎠Q T ⎢=⎣∥Fλ 10.0⎤⎥⎦− δ(Λ)∥2(1504), λ 1 > 0
C.2. DIAGONAL, TRACE, SINGULAR AND EIGEN VALUES 555C.2.0.0.1 Exercise. Rank-1 approximation.Given symmetric matrix A∈ S N , prove:v 1 = arg minimize ‖xx T − A‖ 2 Fxsubject to ‖x‖ = 1(1505)where v 1 is a normalized eigenvector of A corresponding to its largesteigenvalue.(Ky Fan) For B ∈ S N whose eigenvalues λ(B)∈ R N are arranged innonincreasing order, and for 1≤k ≤N [9,4.1] [158] [150,4.3.18][271,2] [176,2.1]N∑i=N−k+1λ(B) i = infU∈ R N×kU T U=Ik∑λ(B) i = supi=1U∈ R N×kU T U=Itr(UU T B) = minimizeX∈ S N +tr(XB)subject to X ≼ ItrX = k= maximize (k − N)µ + tr(B − Z)µ∈R , Z∈S N +subject to µI + Z ≽ Btr(UU T B) = maximizeX∈ S N +tr(XB)subject to X ≼ ItrX = k(a)(b)(c)= minimize kµ + trZµ∈R , Z∈S N +subject to µI + Z ≽ B(d)(1506)Given ordered diagonalization B = QΛQ T , (A.5.2) then an optimalU for the infimum is U ⋆ = Q(:, N − k+1:N)∈ R N×k whereasU ⋆ = Q(:, 1:k)∈ R N×k for the supremum. In both cases, X ⋆ = U ⋆ U ⋆T .<strong>Optimization</strong> (a) searches the convex hull of the outer product UU Tof all N ×k orthonormal matrices. (2.3.2.0.1)
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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 191 Orion
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LIST OF FIGURES 1559 Epigraph . . .
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LIST OF FIGURES 17120 Four fundamen
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Chapter 1OverviewConvex Optimizatio
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ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
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23Figure 4: This coarsely discretiz
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ases (biorthogonal expansion). We e
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27Figure 7: These bees construct a
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its membership to the EDM cone. The
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31appendicesProvided so as to be mo
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Chapter 2Convex geometryConvexity h
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2.1. CONVEX SET 35Figure 11: A slab
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2.1. CONVEX SET 372.1.6 empty set v
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2.1. CONVEX SET 392.1.7.1 Line inte
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2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
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2.1. CONVEX SET 43This theorem in c
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 53Figure 14: Convex hull
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2.3. HULLS 55Aaffine hull (drawn tr
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2.3. HULLS 57The union of relative
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2.4. HALFSPACE, HYPERPLANE 59of dim
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2.4. HALFSPACE, HYPERPLANE 61H +ay
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2.4. HALFSPACE, HYPERPLANE 63Inters
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2.4. HALFSPACE, HYPERPLANE 65Conver
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2.4. HALFSPACE, HYPERPLANE 67A 1A 2
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2.4. HALFSPACE, HYPERPLANE 69tradit
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2.4. HALFSPACE, HYPERPLANE 71There
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2.5. SUBSPACE REPRESENTATIONS 73nor
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2.5. SUBSPACE REPRESENTATIONS 752.5
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2.6. EXTREME, EXPOSED 77In other wo
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2.6. EXTREME, EXPOSED 792.6.1 Expos
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2.7. CONES 812.6.1.3.1 Definition.
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2.7. CONES 830Figure 26: Boundary o
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2.7. CONES 852.7.2 Convex coneWe ca
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2.7. CONES 87Thus the simplest and
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2.7. CONES 89nomenclature generaliz
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2.8. CONE BOUNDARY 91Proper cone {0
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2.8. CONE BOUNDARY 93the same extre
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2.8. CONE BOUNDARY 95For a proper c
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 121
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2.12. CONVEX POLYHEDRA 127It follow
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2.12. CONVEX POLYHEDRA 129Coefficie
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2.12. CONVEX POLYHEDRA 1312.12.3 Un
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2.12. CONVEX POLYHEDRA 133
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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 185f 1 (x)f 2
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3.1. CONVEX FUNCTION 1873.1.3 norm
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3.1. CONVEX FUNCTION 189n∑π(x) i
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3.1. CONVEX FUNCTION 191This means
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3.1. CONVEX FUNCTION 1933.1.5.3 pos
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3.1. CONVEX FUNCTION 1953.1.6.0.1 E
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3.1. CONVEX FUNCTION 1973.1.7 epigr
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3.1. CONVEX FUNCTION 201Although th
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3.1. CONVEX FUNCTION 203minimize tX
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3.1. CONVEX FUNCTION 2053.1.8 gradi
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3.1. CONVEX FUNCTION 209f(Y )[ ∇f
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3.1. CONVEX FUNCTION 211g is convex
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3.1. CONVEX FUNCTION 213αβα ≥
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3.1. CONVEX FUNCTION 2153.1.11 seco
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 221matrices in the
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3.3. QUASICONVEX 223on R M + . Alth
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Chapter 4Semidefinite programmingPr
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4.1. CONIC PROBLEM 227where K is a
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4.1. CONIC PROBLEM 229C0PΓ 1Γ 2S+
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4.1. CONIC PROBLEM 231faces of S 3
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4.1. CONIC PROBLEM 2334.1.1.3 Previ
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4.2. FRAMEWORK 235Equivalently, pri
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4.2. FRAMEWORK 237is positive semid
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4.2. FRAMEWORK 239Optimal value of
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4.2. FRAMEWORK 2414.2.3.1.1 Example
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4.2. FRAMEWORK 243where δ is the m
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4.2. FRAMEWORK 2454.2.3.1.2 Example
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4.3. RANK REDUCTION 2474.3 Rank red
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4.3. RANK REDUCTION 249A rank-reduc
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4.3. RANK REDUCTION 251(t ⋆ i)
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4.3. RANK REDUCTION 2534.3.3.0.1 Ex
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4.3. RANK REDUCTION 2554.3.3.0.2 Ex
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 2734.
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4.5. CONSTRAINING CARDINALITY 275Th
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4.5. CONSTRAINING CARDINALITY 277wh
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4.5. CONSTRAINING CARDINALITY 279f
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4.5. CONSTRAINING CARDINALITY 281al
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.7. CONVEX ITERATION RANK-1 301whi
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4.7. CONVEX ITERATION RANK-1 303the
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Chapter 5Euclidean Distance MatrixT
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5.2. FIRST METRIC PROPERTIES 307cor
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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 313The collecti
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5.4. EDM DEFINITION 3155.4.2 Gram-f
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5.4. EDM DEFINITION 317D ∈ EDM N
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5.4. EDM DEFINITION 3195.4.2.2.1 Ex
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5.4. EDM DEFINITION 321ten affine e
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5.4. EDM DEFINITION 323spheres:Then
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5.4. EDM DEFINITION 325By eliminati
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5.4. EDM DEFINITION 327whereΦ ij =
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5.4. EDM DEFINITION 3295.4.2.2.5 De
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5.4. EDM DEFINITION 331105ˇx 4ˇx
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5.4. EDM DEFINITION 333corrected by
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5.4. EDM DEFINITION 335by translate
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5.4. EDM DEFINITION 337Crippen & Ha
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5.4. EDM DEFINITION 339where ([√t
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5.4. EDM DEFINITION 341because (A.3
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5.5. INVARIANCE 3435.5.1.0.1 Exampl
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 3515.
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5.7. EMBEDDING IN AFFINE HULL 353Fo
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5.7. EMBEDDING IN AFFINE HULL 3555.
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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 373(ii)
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5.11. EDM INDEFINITENESS 3755.11.1
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5.11. EDM INDEFINITENESS 377(confer
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5.11. EDM INDEFINITENESS 379we have
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5.11. EDM INDEFINITENESS 381For pre
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5.12. LIST RECONSTRUCTION 383where
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5.12. LIST RECONSTRUCTION 385(a)(c)
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5.13. RECONSTRUCTION EXAMPLES 387D
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5.13. RECONSTRUCTION EXAMPLES 389Th
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5.13. RECONSTRUCTION EXAMPLES 391wh
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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 4056.1 Defin
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6.2. POLYHEDRAL BOUNDS 407This cone
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6.3.√EDM CONE IS NOT CONVEX 409N
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6.4. A GEOMETRY OF COMPLETION 4116.
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6.4. A GEOMETRY OF COMPLETION 413(a
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6.4. A GEOMETRY OF COMPLETION 415Fi
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6.5. EDM DEFINITION IN 11 T 417and
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6.5. EDM DEFINITION IN 11 T 419then
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6.5. EDM DEFINITION IN 11 T 4216.5.
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6.5. EDM DEFINITION IN 11 T 423D =
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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 435When the Fins
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6.8. DUAL EDM CONE 437Proof. First,
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6.8. DUAL EDM CONE 439EDM 2 = S 2 h
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6.8. DUAL EDM CONE 441whose veracit
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6.8. DUAL EDM CONE 4436.8.1.3.1 Exe
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6.8. DUAL EDM CONE 445has dual affi
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6.8. DUAL EDM CONE 4476.8.1.7 Schoe
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6.9. THEOREM OF THE ALTERNATIVE 449
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6.10. POSTSCRIPT 451When D is an ED
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Chapter 7Proximity problemsIn summa
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In contrast, order of projection on
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457HS N h0EDM NK = S N h ∩ R N×N
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4597.0.3 Problem approachProblems t
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7.1. FIRST PREVALENT PROBLEM: 461fi
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7.1. FIRST PREVALENT PROBLEM: 4637.
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7.1. FIRST PREVALENT PROBLEM: 465di
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7.1. FIRST PREVALENT PROBLEM: 4677.
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7.1. FIRST PREVALENT PROBLEM: 469wh
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7.1. FIRST PREVALENT PROBLEM: 471Th
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7.2. SECOND PREVALENT PROBLEM: 473O
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7.2. SECOND PREVALENT PROBLEM: 475S
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7.2. SECOND PREVALENT PROBLEM: 477r
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7.2. SECOND PREVALENT PROBLEM: 479c
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7.2. SECOND PREVALENT PROBLEM: 4817
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7.3. THIRD PREVALENT PROBLEM: 483fo
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7.3. THIRD PREVALENT PROBLEM: 485a
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7.3. THIRD PREVALENT PROBLEM: 4877.
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7.3. THIRD PREVALENT PROBLEM: 4897.
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7.3. THIRD PREVALENT PROBLEM: 491Ou
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7.4. CONCLUSION 493The rank constra
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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.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
Page 503 and 504: A.3. PROPER STATEMENTS 503A.3.0.0.1
Page 505 and 506: A.3. PROPER STATEMENTS 505By simila
Page 507 and 508: A.3. PROPER STATEMENTS 507Because R
Page 509 and 510: For A,B ∈ R n×n x T Ax ≥ x T B
Page 511 and 512: A.3. PROPER STATEMENTS 511A.3.1.0.2
Page 513 and 514: A.3. PROPER STATEMENTS 513We can de
Page 515 and 516: A.4. SCHUR COMPLEMENT 515Origin of
Page 517 and 518: A.4. SCHUR COMPLEMENT 517Schur-form
Page 519 and 520: A.5. EIGEN DECOMPOSITION 519A.5.0.1
Page 521 and 522: A.6. SINGULAR VALUE DECOMPOSITION,
Page 527 and 528: A.7. ZEROS 527For diagonalizable ma
Page 529 and 530: A.7. ZEROS 529A.7.4For X,A∈ S M +
Page 531 and 532: A.7. ZEROS 531A.7.5.0.1 Proposition
Page 533 and 534: Appendix BSimple matricesMathematic
Page 535 and 536: B.1. RANK-ONE MATRIX (DYAD) 535R(v)
Page 537 and 538: B.1. RANK-ONE MATRIX (DYAD) 537rang
Page 539 and 540: B.2. DOUBLET 539R([u v ])R(Π)= R([
Page 541 and 542: B.3. ELEMENTARY MATRIX 541If λ ≠
Page 543 and 544: B.4. AUXILIARY V -MATRICES 543the n
Page 545 and 546: B.4. AUXILIARY V -MATRICES 54518. V
Page 547 and 548: B.5. ORTHOGONAL MATRIX 547B.5 Ortho
Page 549 and 550: B.5. ORTHOGONAL MATRIX 549B.5.3.0.1
Page 551 and 552: Appendix CSome analytical optimal r
Page 553: C.2. DIAGONAL, TRACE, SINGULAR AND
Page 557 and 558: C.2. DIAGONAL, TRACE, SINGULAR AND
Page 559 and 560: C.3. ORTHOGONAL PROCRUSTES PROBLEM
Page 561 and 562: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 563 and 564: C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
Page 565 and 566: Appendix DMatrix calculusFrom too m
Page 567 and 568: D.1. DIRECTIONAL DERIVATIVE, TAYLOR
Page 587 and 588: D.2. TABLES OF GRADIENTS AND DERIVA
Page 595 and 596: Appendix EProjectionFor any A∈ R
Page 597 and 598: 597Equivalent to the corresponding
Page 599 and 600: E.1. IDEMPOTENT MATRICES 599where R
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Page 603 and 604: E.2. I − P , PROJECTION ON ALGEBR
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.5. PROJECTION EXAMPLES 611E.5.0.0
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E.5. PROJECTION EXAMPLES 613of rela
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E.5. PROJECTION EXAMPLES 615E.5.0.0
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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.9. PROJECTION ON CONVEX SET 629E.
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E.9. PROJECTION ON CONVEX SET 631
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E.9. PROJECTION ON CONVEX SET 633Pr
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E.9. PROJECTION ON CONVEX SET 637wh
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E.9. PROJECTION ON CONVEX SET 639Un
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E.10. ALTERNATING PROJECTION 643bC
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E.10. ALTERNATING PROJECTION 6450
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E.10. ALTERNATING PROJECTION 647E.1
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E.10. ALTERNATING PROJECTION 649y 2
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E.10. ALTERNATING PROJECTION 651By
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E.10. ALTERNATING PROJECTION 653Bar
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E.10. ALTERNATING PROJECTION 655bH
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E.10. ALTERNATING PROJECTION 657K
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E.10. ALTERNATING PROJECTION 659Whe
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Appendix FMatlab programsMade by Th
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F.1. ISEDM() 663% is nonnegativeif
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F.1. ISEDM() 665F.1.1Subroutines fo
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F.2. CONIC INDEPENDENCE, CONICI() 6
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F.2. CONIC INDEPENDENCE, CONICI() 6
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F.3. MAP OF THE USA 671% plot origi
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F.3. MAP OF THE USA 673statelat = d
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F.4. RANK REDUCTION SUBROUTINE, RRF
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F.4. RANK REDUCTION SUBROUTINE, RRF
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F.5. STURM’S PROCEDURE 679F.5 Stu
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F.6. CONVEX ITERATION DEMONSTRATION
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F.6. CONVEX ITERATION DEMONSTRATION
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F.7. FAST MAX CUT 685endoldtrace =
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F.8. SIGNAL DROPOUT PROBLEM 687whil
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Appendix GNotation and a few defini
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691l.i.w.r.tlinearly independentwit
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693t → 0 +t goes to 0 from above;
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695ψ(Z)DDD T (X)D(X) TD −1 (X)D(
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697R n −or R n×n−S nS n⊥S n
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699∂H −∂H +da supporting hype
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701maximizexargsup Xarg supf(x)subj
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703⊁≥not positive definitescala
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Bibliography[1] Suliman Al-Homidan
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BIBLIOGRAPHY 707[16] Keith Ball. An
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BIBLIOGRAPHY 709[38] A. W. Bojanczy
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BIBLIOGRAPHY 711[57] Steven Chu. Au
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BIBLIOGRAPHY 713[76] Frank R. Deuts
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BIBLIOGRAPHY 715(AACC), June 2004.h
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BIBLIOGRAPHY 717[114] John Clifford
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BIBLIOGRAPHY 719[135] Uwe Helmke an
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BIBLIOGRAPHY 721[157] Joakim Jaldé
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BIBLIOGRAPHY 723[178] Jung Rye Lee.
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BIBLIOGRAPHY 725[200] T. S. Motzkin
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BIBLIOGRAPHY 727[220] Teemu Pennane
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BIBLIOGRAPHY 729[240] Joshua A. Sin
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BIBLIOGRAPHY 731[264] Warren S. Tor
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BIBLIOGRAPHY 733[285] Bernard Widro
Page 735 and 736:
Index0-norm, 241, 273, 275, 2941-no
Page 737 and 738:
INDEX 737clipping, 190, 454, 638, 6
Page 739 and 740:
INDEX 739hull, 53, 56, 306of outer
Page 741 and 742:
INDEX 741unique, 519elbow, 457, 458
Page 743 and 744:
INDEX 743polarity, 62handoff, 335ov
Page 745 and 746:
INDEX 745manifold, 23, 24, 105, 410
Page 747 and 748:
INDEX 747objectivebilinear, 256, 27
Page 749 and 750:
INDEX 749Hadamard, 46, 472, 571, 69
Page 751 and 752:
INDEX 751trace, 477, 478heuristic,
Page 753 and 754:
INDEX 753symmetric hollow, 51tangen
Page 755 and 756:
INDEX 755matrix, 65slack, 227vec, 4
Page 757 and 758:
757
Page 760:
Convex Optimization & Euclidean Dis
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v2007.11.26 - Convex Optimization

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