v2007.11.26 - Convex Optimization

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514 APPENDIX A. LINEAR ALGEBRAA.4 Schur complementConsider the Schur-form partitioned matrix G :C T = C , then [44][ ] A BG =B T ≽ 0C⇔ A ≽ 0, B T (I −AA † ) = 0, C −B T A † B ≽ 0⇔ C ≽ 0, B(I −CC † ) = 0, A−BC † B T ≽ 0Given A T = A and(1337)where A † denotes the Moore-Penrose (pseudo)inverse (E). In the firstinstance, I − AA † is a symmetric projection matrix orthogonally projectingon N(A T ). (1704) It is apparently requiredR(B) ⊥ N(A T ) (1338)which precludes A = 0 when B is any nonzero matrix. Note that A ≻ 0 ⇒A † =A −1 ; thereby, the projection matrix vanishes. Likewise, in the secondinstance, I − CC † projects orthogonally on N(C T ). It is requiredR(B T ) ⊥ N(C T ) (1339)which precludes C =0 for B nonzero. Again, C ≻ 0 ⇒ C † = C −1 . So weget, for A or C nonsingular,[ ] A BG =B T ≽ 0C⇔A ≻ 0, C −B T A −1 B ≽ 0orC ≻ 0, A−BC −1 B T ≽ 0(1340)When A is full-rank then, for all B of compatible dimension, R(B) is inR(A). Likewise, when C is full-rank, R(B T ) is in R(C). Thus the flavor,for A and C nonsingular,[ ] A BG =B T ≻ 0C⇔ A ≻ 0, C −B T A −1 B ≻ 0⇔ C ≻ 0, A−BC −1 B T ≻ 0(1341)where C − B T A −1 B is called the Schur complement of A in G , while theSchur complement of C in G is A − BC −1 B T . [103,4.8]
A.4. SCHUR COMPLEMENT 515Origin of the term Schur complement is from complementary inertia:[77,2.4.4] Defineinertia ( G∈ S M) ∆ = {p,z,n} (1342)where p,z,n respectively represent number of positive, zero, and negativeeigenvalues of G ; id est,M = p + z + n (1343)Then, when A is invertible,and when C is invertible,inertia(G) = inertia(A) + inertia(C − B T A −1 B) (1344)inertia(G) = inertia(C) + inertia(A − BC −1 B T ) (1345)When A=C =0, denoting by σ(B)∈ R m + the nonincreasingly orderedsingular values of matrix B ∈ R m×m , then we have the eigenvalues[41,1.2, prob.17]([ ]) [ ]0 B σ(B)λ(G) = λB T =(1346)0 −Ξσ(B)andinertia(G) = inertia(B T B) + inertia(−B T B) (1347)where Ξ is the order-reversing permutation matrix defined in (1533).A.4.0.0.1 Example. Nonnegative polynomial. [27, p.163]Schur-form positive semidefiniteness is necessary and sufficient for quadraticpolynomial nonnegativity; videlicet, for all compatible x[x T 1] [ A bb T c] [ x1]≥ 0 ⇔ x T Ax + 2b T x + c ≥ 0 (1348)A.4.0.0.2 Example. Sparse Schur conditions.Setting matrix A to the identity simplifies the Schur conditions. Oneconsequence relates the definiteness of three quantities:[ ][ ]I BI 0B T ≽ 0 ⇔ C − B T B ≽ 0 ⇔C0 T C −B T ≽ 0 (1349)B
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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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Page 465 and 466: 7.1. FIRST PREVALENT PROBLEM: 465di
Page 467 and 468: 7.1. FIRST PREVALENT PROBLEM: 4677.
Page 469 and 470: 7.1. FIRST PREVALENT PROBLEM: 469wh
Page 471 and 472: 7.1. FIRST PREVALENT PROBLEM: 471Th
Page 473 and 474: 7.2. SECOND PREVALENT PROBLEM: 473O
Page 475 and 476: 7.2. SECOND PREVALENT PROBLEM: 475S
Page 477 and 478: 7.2. SECOND PREVALENT PROBLEM: 477r
Page 479 and 480: 7.2. SECOND PREVALENT PROBLEM: 479c
Page 481 and 482: 7.2. SECOND PREVALENT PROBLEM: 4817
Page 483 and 484: 7.3. THIRD PREVALENT PROBLEM: 483fo
Page 485 and 486: 7.3. THIRD PREVALENT PROBLEM: 485a
Page 487 and 488: 7.3. THIRD PREVALENT PROBLEM: 4877.
Page 489 and 490: 7.3. THIRD PREVALENT PROBLEM: 4897.
Page 491 and 492: 7.3. THIRD PREVALENT PROBLEM: 491Ou
Page 493 and 494: 7.4. CONCLUSION 493The rank constra
Page 495 and 496: Appendix ALinear algebraA.1 Main-di
Page 497 and 498: A.1. MAIN-DIAGONAL δ OPERATOR, λ
Page 499 and 500: A.2. SEMIDEFINITENESS: DOMAIN OF TE
Page 501 and 502: 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: A.3. PROPER STATEMENTS 513We can de
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
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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 and 554: 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
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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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597Equivalent to the corresponding
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E.1. IDEMPOTENT MATRICES 599where R
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E.1. IDEMPOTENT MATRICES 601TxT ⊥
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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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