v2010.10.26 - Convex Optimization

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752 APPENDIX E. PROJECTIONa(a){y | (b −P 1 b) T (y −P 1 b)=0}b(b)C 1yC 1C 1P 1 bC 2C 2bC 2PbP 1 P 2 b(c)Figure 169:(a) (distance) Intersection of two convex sets in R 2 is empty. Method ofalternating projection would be applied to find that point in C 1 nearest C 2 .(b) (distance) Given b ∈ C 2 , then P 1 b ∈ C 1 is nearest b iff(y −P 1 b) T (b −P 1 b)≤0 ∀y ∈ C 1 by the unique minimum-distance projectiontheorem (E.9.1.0.2). When P 1 b attains the distance between the two sets,hyperplane {y | (b −P 1 b) T (y −P 1 b)=0} separates C 1 from C 2 . [61,2.5.1](c) (0 distance) Intersection is nonempty.(optimization) We may want the point Pb in ⋂ C k nearest point b .(feasibility) We may instead be satisfied with a fixed point of the projectionproduct P 1 P 2 b in ⋂ C k .
E.10. ALTERNATING PROJECTION 753E.10.1Distance and existenceExistence of a fixed point is established:E.10.1.0.1 Theorem. Distance. [78]Given any two closed convex sets C 1 and C 2 in R n , then P 1 b∈ C 1 is afixed point of projection product P 1 P 2 if and only if P 1 b is a point of C 1nearest C 2 .⋄Proof. (⇒) Given fixed point a =P 1 P 2 a∈ C 1 with b P 2 a∈ C 2 intandem so that a =P 1 b , then by the unique minimum-distance projectiontheorem (E.9.1.0.2)(2054)(b − a) T (u − a) ≤ 0 ∀u∈ C 1(a − b) T (v − b) ≤ 0⇔∀v ∈ C 2‖a − b‖ ≤ ‖u − v‖ ∀u∈ C 1 and ∀v ∈ C 2by Cauchy-Schwarz inequality [307]|〈x,y〉| ≤ ‖x‖ ‖y‖ (2055)(with equality iff x=κy where κ∈ R [227, p.137]).(⇐) Suppose a∈ C 1 and ‖a − P 2 a‖ ≤ ‖u − P 2 u‖ ∀u∈ C 1 and we chooseu =P 1 P 2 a . Then‖u − P 2 u‖ = ‖P 1 P 2 a − P 2 P 1 P 2 a‖ ≤ ‖a − P 2 a‖ ⇔ a = P 1 P 2 a (2056)Thus a = P 1 b (with b =P 2 a∈ C 2 ) is a fixed point in C 1 of the projectionproduct P 1 P 2 . E.19E.10.2Feasibility and convergenceThe set of all fixed points of any nonexpansive mapping is a closed convexset. [156, lem.3.4] [28,1] The projection product P 1 P 2 is nonexpansive byTheorem E.9.3.0.1 because, for any vectors x,a∈ R n‖P 1 P 2 x − P 1 P 2 a‖ ≤ ‖P 2 x − P 2 a‖ ≤ ‖x − a‖ (2057)E.19 Point b=P 2 a can be shown, similarly, to be a fixed point of product P 2 P 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 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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7.2. SECOND PREVALENT PROBLEM: 573w
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7.2. SECOND PREVALENT PROBLEM: 5757
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7.2. SECOND PREVALENT PROBLEM: 577a
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7.3. THIRD PREVALENT PROBLEM: 579is
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7.3. THIRD PREVALENT PROBLEM: 581We
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7.3. THIRD PREVALENT PROBLEM: 583su
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7.3. THIRD PREVALENT PROBLEM: 585Gi
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7.3. THIRD PREVALENT PROBLEM: 587Op
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7.4. CONCLUSION 589filtering, multi
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Appendix ALinear algebraA.1 Main-di
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.1. MAIN-DIAGONAL δ OPERATOR, λ
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A.2. SEMIDEFINITENESS: DOMAIN OF TE
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A.3. PROPER STATEMENTS 599(AB) T
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A.3. PROPER STATEMENTS 601A.3.1Semi
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A.3. PROPER STATEMENTS 603For A dia
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A.3. PROPER STATEMENTS 605Diagonali
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A.3. PROPER STATEMENTS 607For A,B
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A.3. PROPER STATEMENTS 609When B is
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A.4. SCHUR COMPLEMENT 611A.4 Schur
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A.4. SCHUR COMPLEMENT 613A.4.0.0.3
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A.4. SCHUR COMPLEMENT 615From Corol
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A.5. EIGENVALUE DECOMPOSITION 617wh
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A.5. EIGENVALUE DECOMPOSITION 619A.
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.6. SINGULAR VALUE DECOMPOSITION,
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A.7. ZEROS 625A.6.5SVD of symmetric
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A.7. ZEROS 627(Transpose.)Likewise,
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A.7. ZEROS 629For X,A∈ S M +[34,2
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A.7. ZEROS 631A.7.5.0.1 Proposition
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Appendix BSimple matricesMathematic
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B.1. RANK-ONE MATRIX (DYAD) 635R(v)
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B.1. RANK-ONE MATRIX (DYAD) 637B.1.
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B.2. DOUBLET 639R([u v ])R(Π)= R([
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B.3. ELEMENTARY MATRIX 641has N −
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B.4. AUXILIARY V -MATRICES 643is an
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B.4. AUXILIARY V -MATRICES 64514. [
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B.5. ORTHOGONAL MATRIX 647Given X
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B.5. ORTHOGONAL MATRIX 649Figure 15
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B.5. ORTHOGONAL MATRIX 651which is
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Appendix CSome analytical optimal r
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C.2. TRACE, SINGULAR AND EIGEN VALU
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C.3. ORTHOGONAL PROCRUSTES PROBLEM
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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Appendix DMatrix calculusFrom too m
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D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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D.2. TABLES OF GRADIENTS AND DERIVA
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Appendix EProjectionFor any A∈ R
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Page 779 and 780: 779SDPSVDSNRdBEDMS n 1S n hS n⊥hS
Page 781 and 782: 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
Page 789 and 790: 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.
Page 801 and 802: BIBLIOGRAPHY 801[182] Johan Håstad
Page 803 and 804:
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
Page 811 and 812:
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,
Page 819 and 820:
INDEX 819bees, 30, 413Bellman, 276b
Page 821 and 822:
INDEX 821circular, 130construction,
Page 823 and 824:
INDEX 823measure, 295convex, 21, 36
Page 825 and 826:
INDEX 825duality, 160, 410gap, 161,
Page 827 and 828:
INDEX 827positive semidefinite, 284
Page 829 and 830:
INDEX 829point, 39, 62positive semi
Page 831 and 832:
INDEX 831isomorphic, 52, 56, 59, 77
Page 833 and 834:
INDEX 833nonsymmetric, 703range, 70
Page 835 and 836:
INDEX 835Muller, 624multidimensiona
Page 837 and 838:
INDEX 837projection on, 739, 745tra
Page 839 and 840:
INDEX 839pseudoinverse, 701trace, 5
Page 841 and 842:
INDEX 841coordinate system, 156line
Page 843 and 844:
INDEX 843nonconvex, 40, 97-101nulls
Page 845 and 846:
INDEX 845faces, 106intersection, 10
Page 847 and 848:
INDEX 847projection, 517symmetric,
Page 849 and 850:
849
Page 852:
Convex Optimization & Euclidean Dis
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