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810 BIBLIOGRAPHY[304] R. Tyrrell Rockafellar. Conjugate Duality and Optimization. SIAM, 1974.[305] R. Tyrrell Rockafellar. Lagrange multipliers in optimization. SIAM-AmericanMathematical Society Proceedings, 9:145–168, 1976.[306] R. Tyrrell Rockafellar. Lagrange multipliers and optimality. SIAM Review,35(2):183–238, June 1993.[307] R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1997. Firstpublished in 1970.[308] C. K. Rushforth. Signal restoration, functional analysis, and Fredholm integralequations of the first kind. In Henry Stark, editor, Image Recovery: Theory andApplication, chapter 1, pages 1–27. Academic Press, 1987.[309] Peter Santos. Application platforms and synthesizable analog IP. In Proceedings ofthe Embedded Systems Conference, San Francisco California USA, 2003.http://www.convexoptimization.com/TOOLS/Santos.pdf[310] Shankar Sastry. Nonlinear Systems: Analysis, Stability, and Control.Springer-Verlag, 1999.[311] Uwe Schäfer. A linear complementarity problem with a P-matrix. SIAM Review,46(2):189–201, June 2004.[312] Isaac J. Schoenberg. Remarks to Maurice Fréchet’s article “Sur ladéfinition axiomatique d’une classe d’espace distanciés vectoriellement applicablesur l’espace de Hilbert”. Annals of Mathematics, 36(3):724–732, July 1935.http://www.convexoptimization.com/TOOLS/Schoenberg2.pdf[313] Isaac J. Schoenberg. Metric spaces and positive definite functions. Transactions ofthe American Mathematical Society, 44:522–536, 1938.http://www.convexoptimization.com/TOOLS/Schoenberg3.pdf[314] Peter H. Schönemann. A generalized solution of the orthogonal Procrustes problem.Psychometrika, 31(1):1–10, 1966.[315] Peter H. Schönemann, Tim Dorcey, and K. Kienapple. Subadditive concatenationin dissimilarity judgements. Perception and Psychophysics, 38:1–17, 1985.[316] Alexander Schrijver. On the history of combinatorial optimization (till 1960). InK. Aardal, G. L. Nemhauser, and R. Weismantel, editors, Handbook of DiscreteOptimization, pages 1–68. Elsevier, 2005.http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.3362[317] Seymour Sherman. Doubly stochastic matrices and complex vector spaces.American Journal of Mathematics, 77(2):245–246, April 1955.http://www.convexoptimization.com/TOOLS/Sherman.pdf[318] Joshua A. Singer. Log-Penalized Linear Regression. PhD thesis, Stanford University,Department of Electrical Engineering, June 2004.http://www.convexoptimization.com/TOOLS/Josh.pdf
BIBLIOGRAPHY 811[319] Anthony Man-Cho So and Yinyu Ye. A semidefinite programming approach totensegrity theory and realizability of graphs. In Proceedings of the SeventeenthAnnual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 766–775,Miami Florida USA, January 2006. Association for Computing Machinery (ACM).http://www.convexoptimization.com/TOOLS/YeSo.pdf[320] Anthony Man-Cho So and Yinyu Ye. Theory of semidefinite programming for sensornetwork localization. Mathematical Programming: Series A and B, 109(2):367–384,January 2007.http://www.stanford.edu/~yyye/local-theory.pdf[321] D. C. Sorensen. Newton’s method with a model trust region modification. SIAMJournal on Numerical Analysis, 19(2):409–426, April 1982.http://www.convexoptimization.com/TOOLS/soren.pdf[322] Steve Spain. Polaris Wireless corp., 2005.http://www.polariswireless.comPersonal communication.[323] Nathan Srebro, Jason D. M. Rennie, and Tommi S. Jaakkola. Maximum-marginmatrix factorization. In Lawrence K. Saul, Yair Weiss, and Léon Bottou,editors, Advances in Neural Information Processing Systems 17 (NIPS 2004), pages1329–1336. MIT Press, 2005.[324] Wolfram Stadler, editor. Multicriteria Optimization in Engineering and in theSciences. Springer-Verlag, 1988.[325] Henry Stark, editor. Image Recovery: Theory and Application. Academic Press,1987.[326] Henry Stark. Polar, spiral, and generalized sampling and interpolation. In RobertJ. Marks II, editor, Advanced Topics in Shannon Sampling and Interpolation Theory,chapter 6, pages 185–218. Springer-Verlag, 1993.[327] Willi-Hans Steeb. Matrix Calculus and Kronecker Product with Applications andC++ Programs. World Scientific, 1997.[328] Gilbert W. Stewart and Ji-guang Sun. Matrix Perturbation Theory. Academic Press,1990.[329] Josef Stoer. On the characterization of least upper bound norms in matrix space.Numerische Mathematik, 6(1):302–314, December 1964.http://www.convexoptimization.com/TOOLS/Stoer.pdf[330] Josef Stoer and Christoph Witzgall. Convexity and Optimization in FiniteDimensions I. Springer-Verlag, 1970.[331] Gilbert Strang. Linear Algebra and its Applications. Harcourt Brace, third edition,1988.[332] Gilbert Strang. Calculus. Wellesley-Cambridge Press, 1992.
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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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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
Page 735 and 736:
E.9. PROJECTION ON CONVEX SET 735As
Page 737 and 738:
E.9. PROJECTION ON CONVEX SET 737Wi
Page 739 and 740:
E.9. PROJECTION ON CONVEX SET 739R(
Page 741 and 742:
E.9. PROJECTION ON CONVEX SET 741E.
Page 743 and 744:
E.9. PROJECTION ON CONVEX SET 743E.
Page 745 and 746:
E.9. PROJECTION ON CONVEX SET 745Un
Page 747 and 748:
E.9. PROJECTION ON CONVEX SET 747ac
Page 749 and 750:
E.10. ALTERNATING PROJECTION 749bC
Page 751 and 752:
E.10. ALTERNATING PROJECTION 7510
Page 753 and 754:
E.10. ALTERNATING PROJECTION 753E.1
Page 755 and 756:
E.10. ALTERNATING PROJECTION 755y 2
Page 757 and 758:
E.10. ALTERNATING PROJECTION 757Def
Page 759 and 760: E.10. ALTERNATING PROJECTION 759Dis
Page 761 and 762: E.10. ALTERNATING PROJECTION 761mat
Page 763 and 764: E.10. ALTERNATING PROJECTION 763K
Page 765 and 766: E.10. ALTERNATING PROJECTION 765E.1
Page 767 and 768: E.10. ALTERNATING PROJECTION 767E.1
Page 769 and 770: Appendix FNotation and a few defini
Page 771 and 772: 771A ij or A(i, j) , ij th entry of
Page 773 and 774: 773⊞orthogonal vector sum of sets
Page 775 and 776: 775x +vector x whose negative entri
Page 777 and 778: 777X point list ((76) having cardin
Page 779 and 780: 779SDPSVDSNRdBEDMS n 1S n hS n⊥hS
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: BIBLIOGRAPHY 809Notes in Computer S
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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