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796 BIBLIOGRAPHY[116] Bruce Randall Donald. 3-D structure in chemistry and molecular biology, 1998.http://www.cs.duke.edu/brd/Teaching/Previous/Bio[117] David L. Donoho. Neighborly polytopes and sparse solution of underdeterminedlinear equations. Technical Report 2005-04, Stanford University, Department ofStatistics, January 2005.www-stat.stanford.edu/~donoho/Reports/2005/NPaSSULE-01-28-05.pdf[118] David L. Donoho. Compressed sensing. IEEE Transactions on Information Theory,52(4):1289–1306, April 2006.www-stat.stanford.edu/~donoho/Reports/2004/CompressedSensing091604.pdf[119] David L. Donoho. High-dimensional centrally symmetric polytopes withneighborliness proportional to dimension. Discrete & Computational Geometry,35(4):617–652, May 2006.[120] David L. Donoho, Michael Elad, and Vladimir Temlyakov. Stable recovery ofsparse overcomplete representations in the presence of noise. IEEE Transactionson Information Theory, 52(1):6–18, January 2006.www-stat.stanford.edu/~donoho/Reports/2004/StableSparse-Donoho-etal.pdf[121] David L. Donoho and Philip B. Stark. Uncertainty principles and signal recovery.SIAM Journal on Applied Mathematics, 49(3):906–931, June 1989.[122] David L. Donoho and Jared Tanner. Neighborliness of randomly projectedsimplices in high dimensions. Proceedings of the National Academy of Sciences,102(27):9452–9457, July 2005.[123] David L. Donoho and Jared Tanner. Sparse nonnegative solution of underdeterminedlinear equations by linear programming. Proceedings of the National Academy ofSciences, 102(27):9446–9451, July 2005.[124] David L. Donoho and Jared Tanner. Counting faces of randomly projectedpolytopes when the projection radically lowers dimension. Journal of the AmericanMathematical Society, 22(1):1–53, January 2009.[125] Miguel Nuno Ferreira Fialho dos Anjos. New Convex Relaxations for the MaximumCut and VLSI Layout Problems. PhD thesis, University of Waterloo, OntarioCanada, Department of Combinatorics and Optimization, 2001.http://etd.uwaterloo.ca/etd/manjos2001.pdf[126] John Duchi, Shai Shalev-Shwartz, Yoram Singer, and Tushar Chandra. Efficientprojections onto the l 1 -ball for learning in high dimensions. In Proceedings ofthe 25 th International Conference on Machine Learning (ICML), pages 272–279,Helsinki Finland, July 2008. Association for Computing Machinery (ACM).http://icml2008.cs.helsinki.fi/papers/361.pdf[127] Richard L. Dykstra. An algorithm for restricted least squares regression. Journal ofthe American Statistical Association, 78(384):837–842, 1983.[128] Peter J. Eccles. An Introduction to Mathematical Reasoning: numbers, sets andfunctions. Cambridge University Press, 1997.
BIBLIOGRAPHY 797[129] Carl Eckart and Gale Young. The approximation of one matrix by another of lowerrank. Psychometrika, 1(3):211–218, September 1936.www.convexoptimization.com/TOOLS/eckart&young.1936.pdf[130] Alan Edelman, Tomás A. Arias, and Steven T. Smith. The geometry of algorithmswith orthogonality constraints. SIAM Journal on Matrix Analysis and Applications,20(2):303–353, 1998.[131] John J. Edgell. Graphics calculator applications on 4-D constructs, 1996.http://archives.math.utk.edu/ICTCM/EP-9/C47/pdf/paper.pdf[132] Ivar Ekeland and Roger Témam. Convex Analysis and Variational Problems. SIAM,1999.[133] Julius Farkas. Theorie der einfachen Ungleichungen. Journal für die reine undangewandte Mathematik, 124:1–27, 1902.http://gdz.sub.uni-goettingen.de/no cache/dms/load/img/?IDDOC=261361[134] Maryam Fazel. Matrix Rank Minimization with Applications. PhD thesis, StanfordUniversity, Department of Electrical Engineering, March 2002.http://www.cds.caltech.edu/~maryam/thesis-final.pdf[135] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. A rank minimization heuristicwith application to minimum order system approximation. In Proceedings of theAmerican Control Conference, volume 6, pages 4734–4739. American AutomaticControl Council (AACC), June 2001.http://www.cds.caltech.edu/~maryam/nucnorm.html[136] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. Log-det heuristic formatrix rank minimization with applications to Hankel and Euclidean distancematrices. In Proceedings of the American Control Conference, volume 3,pages 2156–2162. American Automatic Control Council (AACC), June 2003.http://www.cds.caltech.edu/~maryam/acc03 final.pdf[137] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. Rank minimization andapplications in system theory. In Proceedings of the American Control Conference,volume 4, pages 3273–3278. American Automatic Control Council (AACC), June2004.http://www.cds.caltech.edu/~maryam/acc04-tutorial.pdf[138] J. T. Feddema, R. H. Byrne, J. J. Harrington, D. M. Kilman, C. L. Lewis,R. D. Robinett, B. P. Van Leeuwen, and J. G. Young. Advanced mobilenetworking, sensing, and controls. Sandia Report SAND2005-1661, Sandia NationalLaboratories, Albuquerque New Mexico USA, March 2005.www.prod.sandia.gov/cgi-bin/techlib/access-control.pl/2005/051661.pdf[139] César Fernández, Thomas Szyperski, Thierry Bruyère, Paul Ramage, EgonMösinger, and Kurt Wüthrich. NMR solution structure of the pathogenesis-relatedprotein P14a. Journal of Molecular Biology, 266:576–593, 1997.
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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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2.2. VECTORIZED-MATRIX INNER PRODUC
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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 792.4.2.
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2.4. HALFSPACE, HYPERPLANE 81tradit
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2.4. HALFSPACE, HYPERPLANE 832.4.2.
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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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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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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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2.10. CONIC INDEPENDENCE (C.I.) 143
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2.10. CONIC INDEPENDENCE (C.I.) 145
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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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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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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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3.2. PRACTICAL NORM FUNCTIONS, ABSO
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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 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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3.7. CONVEX MATRIX-VALUED FUNCTION
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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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4.7. CONSTRAINING RANK OF INDEFINIT
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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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5.6. 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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5.8. EUCLIDEAN METRIC VERSUS MATRIX
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5.9. BRIDGE: CONVEX POLYHEDRA TO ED
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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 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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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Page 495 and 496:
5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Page 497 and 498:
Chapter 6Cone of distance matricesF
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Page 499 and 500:
6.1. DEFINING EDM CONE 4996.1 Defin
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Page 501 and 502:
6.2. POLYHEDRAL BOUNDS 501This cone
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Page 503 and 504:
6.4. EDM DEFINITION IN 11 T 503That
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Page 505 and 506:
6.4. EDM DEFINITION IN 11 T 505N(1
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Page 507 and 508:
6.4. EDM DEFINITION IN 11 T 507Then
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Page 509 and 510:
6.4. EDM DEFINITION IN 11 T 5096.4.
-
Page 511 and 512:
6.5. CORRESPONDENCE TO PSD CONE S N
-
Page 513 and 514:
6.5. CORRESPONDENCE TO PSD CONE S N
-
Page 515 and 516:
6.5. CORRESPONDENCE TO PSD CONE S N
-
Page 517 and 518:
6.6. VECTORIZATION & PROJECTION INT
-
Page 519 and 520:
6.6. VECTORIZATION & PROJECTION INT
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Page 521 and 522:
6.6. VECTORIZATION & PROJECTION INT
-
Page 523 and 524:
6.7. A GEOMETRY OF COMPLETION 523(b
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Page 525 and 526:
6.7. A GEOMETRY OF COMPLETION 525[3
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Page 527 and 528:
6.7. A GEOMETRY OF COMPLETION 527wh
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Page 529 and 530:
6.8. DUAL EDM CONE 529to the geomet
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6.8. DUAL EDM CONE 531Proof. First,
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Page 533 and 534:
6.8. DUAL EDM CONE 533EDM 2 = S 2 h
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Page 535 and 536:
6.8. DUAL EDM CONE 535therefore the
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Page 537 and 538:
6.8. DUAL EDM CONE 537Elegance of t
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Page 539 and 540:
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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Page 543 and 544:
6.8. DUAL EDM CONE 5430dvec rel ∂
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Page 545 and 546:
6.10. POSTSCRIPT 5456.10 Postscript
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Page 547 and 548:
Chapter 7Proximity problemsIn the
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Page 549 and 550:
549project on the subspace, then pr
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Page 551 and 552:
551HS N h0EDM NK = S N h ∩ R N×N
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Page 553 and 554:
5537.0.3 Problem approachProblems t
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Page 555 and 556:
7.1. FIRST PREVALENT PROBLEM: 555fi
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Page 557 and 558:
7.1. FIRST PREVALENT PROBLEM: 5577.
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Page 559 and 560:
7.1. FIRST PREVALENT PROBLEM: 559di
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Page 561 and 562:
7.1. FIRST PREVALENT PROBLEM: 5617.
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Page 563 and 564:
7.1. FIRST PREVALENT PROBLEM: 563wh
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Page 565 and 566:
7.1. FIRST PREVALENT PROBLEM: 565Th
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Page 567 and 568:
7.2. SECOND PREVALENT PROBLEM: 567O
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Page 569 and 570:
7.2. SECOND PREVALENT PROBLEM: 569S
-
Page 571 and 572:
7.2. SECOND PREVALENT PROBLEM: 571r
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Page 573 and 574:
7.2. SECOND PREVALENT PROBLEM: 573w
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Page 575 and 576:
7.2. SECOND PREVALENT PROBLEM: 5757
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Page 577 and 578:
7.2. SECOND PREVALENT PROBLEM: 577a
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Page 579 and 580:
7.3. THIRD PREVALENT PROBLEM: 579is
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Page 581 and 582:
7.3. THIRD PREVALENT PROBLEM: 581We
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Page 583 and 584:
7.3. THIRD PREVALENT PROBLEM: 583su
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Page 585 and 586:
7.3. THIRD PREVALENT PROBLEM: 585Gi
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Page 587 and 588:
7.3. THIRD PREVALENT PROBLEM: 587Op
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Page 589 and 590:
7.4. CONCLUSION 589filtering, multi
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Page 591 and 592:
Appendix ALinear algebraA.1 Main-di
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Page 593 and 594:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
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Page 595 and 596:
A.1. MAIN-DIAGONAL δ OPERATOR, λ
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Page 597 and 598:
A.2. SEMIDEFINITENESS: DOMAIN OF TE
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Page 599 and 600:
A.3. PROPER STATEMENTS 599(AB) T
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Page 601 and 602:
A.3. PROPER STATEMENTS 601A.3.1Semi
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Page 603 and 604:
A.3. PROPER STATEMENTS 603For A dia
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Page 605 and 606:
A.3. PROPER STATEMENTS 605Diagonali
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Page 607 and 608:
A.3. PROPER STATEMENTS 607For A,B
-
Page 609 and 610:
A.3. PROPER STATEMENTS 609When B is
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Page 611 and 612:
A.4. SCHUR COMPLEMENT 611A.4 Schur
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Page 613 and 614:
A.4. SCHUR COMPLEMENT 613A.4.0.0.3
-
Page 615 and 616:
A.4. SCHUR COMPLEMENT 615From Corol
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Page 617 and 618:
A.5. EIGENVALUE DECOMPOSITION 617wh
-
Page 619 and 620:
A.5. EIGENVALUE DECOMPOSITION 619A.
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Page 621 and 622:
A.6. SINGULAR VALUE DECOMPOSITION,
-
Page 623 and 624:
A.6. SINGULAR VALUE DECOMPOSITION,
-
Page 625 and 626:
A.7. ZEROS 625A.6.5SVD of symmetric
-
Page 627 and 628:
A.7. ZEROS 627(Transpose.)Likewise,
-
Page 629 and 630:
A.7. ZEROS 629For X,A∈ S M +[34,2
-
Page 631 and 632:
A.7. ZEROS 631A.7.5.0.1 Proposition
-
Page 633 and 634:
Appendix BSimple matricesMathematic
-
Page 635 and 636:
B.1. RANK-ONE MATRIX (DYAD) 635R(v)
-
Page 637 and 638:
B.1. RANK-ONE MATRIX (DYAD) 637B.1.
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Page 639 and 640:
B.2. DOUBLET 639R([u v ])R(Π)= R([
-
Page 641 and 642:
B.3. ELEMENTARY MATRIX 641has N −
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Page 643 and 644:
B.4. AUXILIARY V -MATRICES 643is an
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Page 645 and 646:
B.4. AUXILIARY V -MATRICES 64514. [
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Page 647 and 648:
B.5. ORTHOGONAL MATRIX 647Given X
-
Page 649 and 650:
B.5. ORTHOGONAL MATRIX 649Figure 15
-
Page 651 and 652:
B.5. ORTHOGONAL MATRIX 651which is
-
Page 653 and 654:
Appendix CSome analytical optimal r
-
Page 655 and 656:
C.2. TRACE, SINGULAR AND EIGEN VALU
-
Page 657 and 658:
C.2. TRACE, SINGULAR AND EIGEN VALU
-
Page 659 and 660:
C.2. TRACE, SINGULAR AND EIGEN VALU
-
Page 661 and 662:
C.3. ORTHOGONAL PROCRUSTES PROBLEM
-
Page 663 and 664:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
-
Page 665 and 666:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
-
Page 667 and 668:
C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
-
Page 669 and 670:
Appendix DMatrix calculusFrom too m
-
Page 671 and 672:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
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Page 673 and 674:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 675 and 676:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 677 and 678:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 679 and 680:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 681 and 682:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 683 and 684:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 685 and 686:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 687 and 688:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 689 and 690:
D.1. DIRECTIONAL DERIVATIVE, TAYLOR
-
Page 691 and 692:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 693 and 694:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 695 and 696:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 697 and 698:
D.2. TABLES OF GRADIENTS AND DERIVA
-
Page 699 and 700:
Appendix EProjectionFor any A∈ R
-
Page 701 and 702:
701U T = U † for orthonormal (inc
-
Page 703 and 704:
E.1. IDEMPOTENT MATRICES 703where A
-
Page 705 and 706:
E.1. IDEMPOTENT MATRICES 705order,
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Page 707 and 708:
E.1. IDEMPOTENT MATRICES 707are lin
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Page 709 and 710:
E.3. SYMMETRIC IDEMPOTENT MATRICES
-
Page 711 and 712:
E.3. SYMMETRIC IDEMPOTENT MATRICES
-
Page 713 and 714:
E.3. SYMMETRIC IDEMPOTENT MATRICES
-
Page 715 and 716:
E.4. ALGEBRA OF PROJECTION ON AFFIN
-
Page 717 and 718:
E.5. PROJECTION EXAMPLES 717a ∗ 2
-
Page 719 and 720:
E.5. PROJECTION EXAMPLES 719where Y
-
Page 721 and 722:
E.5. PROJECTION EXAMPLES 721(B.4.2)
-
Page 723 and 724:
E.6. VECTORIZATION INTERPRETATION,
-
Page 725 and 726:
E.6. VECTORIZATION INTERPRETATION,
-
Page 727 and 728:
E.6. VECTORIZATION INTERPRETATION,
-
Page 729 and 730:
E.7. PROJECTION ON MATRIX SUBSPACES
-
Page 731 and 732:
E.7. PROJECTION ON MATRIX SUBSPACES
-
Page 733 and 734:
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:
BIBLIOGRAPHY 795[102] Etienne de Kl
-
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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