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724 BIBLIOGRAPHY [39] Pratik Biswas, Tzu-Chen Liang, Kim-Chuan Toh, Yinyu Ye, and Ta-Chung Wang. Semidefinite programming approaches for sensor network localization with noisy distance measurements. IEEE Transactions on Automation Science and Engineering, 3(4):360–371, October 2006. [40] Pratik Biswas, Tzu-Chen Liang, Ta-Chung Wang, and Yinyu Ye. Semidefinite programming based algorithms for sensor network localization. ACM Transactions on Sensor Networks, 2(2):188–220, May 2006. http://www.stanford.edu/∼yyye/combined rev3.pdf [41] Pratik Biswas, Kim-Chuan Toh, and Yinyu Ye. A distributed SDP approach for large-scale noisy anchor-free graph realization with applications to molecular conformation. SIAM Journal on Scientific Computing, 30(3):1251–1277, March 2008. http://www.stanford.edu/∼yyye/SISC-molecule-revised-2.pdf [42] Pratik Biswas and Yinyu Ye. Semidefinite programming for ad hoc wireless sensor network localization. In Third International Symposium on Information Processing in Sensor Networks (IPSN), pages 46–54. IEEE, April 2004. http://www.stanford.edu/∼yyye/adhocn2.pdf [43] Leonard M. Blumenthal. On the four-point property. Bulletin of the American Mathematical Society, 39:423–426, 1933. [44] Leonard M. Blumenthal. Theory and Applications of Distance Geometry. Oxford University Press, 1953. [45] A. W. Bojanczyk and A. Lutoborski. The Procrustes problem for orthogonal Stiefel matrices. SIAM Journal on Scientific Computing, 21(4):1291–1304, February 1998. Date is electronic publication: http://citeseer.ist.psu.edu/500185.html [46] Ingwer Borg and Patrick Groenen. Modern Multidimensional Scaling. Springer-Verlag, 1997. [47] Jonathan M. Borwein and Heinz Bauschke. Projection algorithms and monotone operators, 1998. http://oldweb.cecm.sfu.ca/personal/jborwein/projections4.pdf [48] Jonathan M. Borwein and Adrian S. Lewis. Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer-Verlag, 2000. http://www.convexoptimization.com/TOOLS/Borwein.pdf [49] Richard Bouldin. The pseudo-inverse of a product. SIAM Journal on Applied Mathematics, 24(4):489–495, June 1973. http://www.convexoptimization.com/TOOLS/Pseudoinverse.pdf [50] Stephen Boyd and Jon Dattorro. Alternating projections, 2003. http://www.stanford.edu/class/ee392o/alt proj.pdf [51] Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994.
BIBLIOGRAPHY 725 [52] Stephen Boyd, Seung-Jean Kim, Lieven Vandenberghe, and Arash Hassibi. A tutorial on geometric programming. Optimization and Engineering, 8(1):1389–4420, March 2007. http://www.stanford.edu/∼boyd/papers/gp tutorial.html [53] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. http://www.stanford.edu/∼boyd/cvxbook [54] James P. Boyle and Richard L. Dykstra. A method for finding projections onto the intersection of convex sets in Hilbert spaces. In R. Dykstra, T. Robertson, and F. T. Wright, editors, Advances in Order Restricted Statistical Inference, pages 28–47. Springer-Verlag, 1986. [55] Lev M. Brègman. The method of successive projection for finding a common point of convex sets. Soviet Mathematics, 162(3):487–490, 1965. AMS translation of Doklady Akademii Nauk SSSR, 6:688-692. [56] Lev M. Brègman, Yair Censor, Simeon Reich, and Yael Zepkowitz-Malachi. Finding the projection of a point onto the intersection of convex sets via projections onto halfspaces, 2003. http://www.optimization-online.org/DB FILE/2003/06/669.pdf [57] Mike Brookes. Matrix reference manual: Matrix calculus, 2002. http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html [58] Richard A. Brualdi. Combinatorial Matrix Classes. Cambridge University Press, 2006. [59] Emmanuel J. Candès. Compressive sampling. In Proceedings of the International Congress of Mathematicians, Madrid, volume III, pages 1433–1452. European Mathematical Society, August 2006. http://www.acm.caltech.edu/∼emmanuel/papers/CompressiveSampling.pdf [60] Emmanuel J. Candès and Justin K. Romberg. l 1 -magic : Recovery of sparse signals via convex programming, October 2005. http://www.acm.caltech.edu/l1magic/downloads/l1magic.pdf [61] Emmanuel J. Candès, Justin K. Romberg, and Terence Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52(2):489–509, February 2006. http://arxiv.org/abs/math.NA/0409186 (2004 DRAFT) http://www.acm.caltech.edu/∼emmanuel/papers/ExactRecovery.pdf [62] Emmanuel J. Candès, Michael B. Wakin, and Stephen P. Boyd. Enhancing sparsity by reweighted l 1 minimization. Journal of Fourier Analysis and Applications, October 2008. http://arxiv.org/abs/0711.1612v1
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DATTORRO CONVEX OPTIMIZATION & EUCL
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Convex Optimization & Euclidean Dis
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for Jennie Columba ♦ Antonio ♦
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Prelude The constant demands of my
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Convex Optimization & Euclidean Dis
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CONVEX OPTIMIZATION & EUCLIDEAN DIS
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List of Figures 1 Overview 19 1 Ori
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LIST OF FIGURES 15 3 Geometry of co
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LIST OF FIGURES 17 126 Decomposing
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Chapter 1 Overview Convex Optimizat
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ˇx 4 ˇx 3 ˇx 2 Figure 2: Applica
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23 Figure 4: This coarsely discreti
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ases (biorthogonal expansion). We e
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27 Figure 7: These bees construct a
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that establish its membership to th
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31 appendices Provided so as to be
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Chapter 2 Convex geometry Convexity
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2.1. CONVEX SET 35 2.1.2 linear ind
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2.1. CONVEX SET 37 2.1.6 empty set
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2.1. CONVEX SET 39 2.1.7.1 Line int
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2.1. CONVEX SET 41 (a) R 2 (b) R 3
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2.1. CONVEX SET 43 This theorem in
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2.2. VECTORIZED-MATRIX INNER PRODUC
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2.3. HULLS 55 Figure 16: Convex hul
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2.3. HULLS 57 The affine hull of tw
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2.3. HULLS 59 2.3.2 Convex hull The
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2.3. HULLS 61 In case k = N , the F
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2.3. HULLS 63 2.3.2.0.3 Exercise. C
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2.3. HULLS 65 Figure 20: A simplici
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2.4. HALFSPACE, HYPERPLANE 67 H + a
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2.4. HALFSPACE, HYPERPLANE 69 1 1
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2.4. HALFSPACE, HYPERPLANE 71 Recal
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2.4. HALFSPACE, HYPERPLANE 73 C H
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2.4. HALFSPACE, HYPERPLANE 75 2.4.2
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2.4. HALFSPACE, HYPERPLANE 77 (conf
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2.5. SUBSPACE REPRESENTATIONS 79 Ra
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2.5. SUBSPACE REPRESENTATIONS 81 If
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2.6. EXTREME, EXPOSED 83 2.6 Extrem
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2.6. EXTREME, EXPOSED 85 A B C D Fi
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2.7. CONES 87 2.6.1.3.1 Definition.
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2.7. CONES 89 0 Figure 30: Boundary
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2.7. CONES 91 2.7.2 Convex cone We
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2.7. CONES 93 Then a pointed closed
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2.7. CONES 95 A pointed closed conv
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2.8. CONE BOUNDARY 97 So the ray th
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2.8. CONE BOUNDARY 99 2.8.1.1 extre
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2.8. CONE BOUNDARY 101 2.8.2 Expose
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2.8. CONE BOUNDARY 103 From Theorem
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2.9. POSITIVE SEMIDEFINITE (PSD) CO
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2.10. CONIC INDEPENDENCE (C.I.) 127
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2.12. CONVEX POLYHEDRA 133 It follo
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2.12. CONVEX POLYHEDRA 135 Coeffici
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2.12. CONVEX POLYHEDRA 137 2.12.3 U
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2.12. CONVEX POLYHEDRA 139
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2.13. DUAL CONE & GENERALIZED INEQU
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Chapter 3 Geometry of convex functi
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3.1. CONVEX FUNCTION 197 f 1 (x) f
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3.1. CONVEX FUNCTION 199 3.1.3 norm
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3.1. CONVEX FUNCTION 201 A B 1 Figu
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3.1. CONVEX FUNCTION 203 k/m 1 0.9
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k∑ i=1 3.1. CONVEX FUNCTION 205 S
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3.1. CONVEX FUNCTION 207 rather x >
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3.1. CONVEX FUNCTION 209 rather ] x
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3.1. CONVEX FUNCTION 211 3.1.6.0.2
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3.1. CONVEX FUNCTION 213 q(x) f(x)
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3.1. CONVEX FUNCTION 215 3.1.7.0.2
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3.1. CONVEX FUNCTION 217 We learned
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3.1. CONVEX FUNCTION 219 Since opti
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3.1. CONVEX FUNCTION 221 2 1.5 1 0.
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3.1. CONVEX FUNCTION 223 Setting th
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3.1. CONVEX FUNCTION 225 Similarly,
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3.1. CONVEX FUNCTION 227 For vector
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3.1. CONVEX FUNCTION 229 This means
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3.1. CONVEX FUNCTION 231 f(Y ) −
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3.2. MATRIX-VALUED CONVEX FUNCTION
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3.3. QUASICONVEX 239 exponential al
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3.3. QUASICONVEX 241 Unlike convex
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3.4. SALIENT PROPERTIES 243 6. (af
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Chapter 4 Semidefinite programming
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4.1. CONIC PROBLEM 247 where K is a
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4.1. CONIC PROBLEM 249 4.1.1.2 Redu
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4.1. CONIC PROBLEM 251 In any SDP f
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4.1. CONIC PROBLEM 253 Proposition
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4.2. FRAMEWORK 255 sets are closed
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4.2. FRAMEWORK 257 4.2.1.1.3 Exampl
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4.2. FRAMEWORK 259 4.2.2 Duals The
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4.2. FRAMEWORK 261 When equality is
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4.2. FRAMEWORK 263 The pseudoinvers
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4.2. FRAMEWORK 265 For the data giv
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4.2. FRAMEWORK 267 minimizes an aff
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4.3. RANK REDUCTION 269 whose rank
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4.3. RANK REDUCTION 271 and where m
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4.3. RANK REDUCTION 273 4.3.3 Optim
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4.3. RANK REDUCTION 275 Initialize:
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4.4. RANK-CONSTRAINED SEMIDEFINITE
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4.5. CONSTRAINING CARDINALITY 295 m
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4.5. CONSTRAINING CARDINALITY 297 m
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4.5. CONSTRAINING CARDINALITY 299 a
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4.5. CONSTRAINING CARDINALITY 301 f
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4.5. CONSTRAINING CARDINALITY 303 n
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4.5. CONSTRAINING CARDINALITY 305 W
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4.5. CONSTRAINING CARDINALITY 307 t
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4.6. CARDINALITY AND RANK CONSTRAIN
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4.7. CONVEX ITERATION RANK-1 341 fi
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4.7. CONVEX ITERATION RANK-1 343 Gi
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Chapter 5 Euclidean Distance Matrix
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5.2. FIRST METRIC PROPERTIES 347 co
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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 353 The collect
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5.4. EDM DEFINITION 355 5.4.2 Gram-
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5.4. EDM DEFINITION 357 D ∈ EDM N
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5.4. EDM DEFINITION 359 5.4.2.2.1 E
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5.4. EDM DEFINITION 361 ten affine
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5.4. EDM DEFINITION 363 spheres: Th
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5.4. EDM DEFINITION 365 By eliminat
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5.4. EDM DEFINITION 367 where Φ ij
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5.4. EDM DEFINITION 369 5.4.2.2.6 D
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5.4. EDM DEFINITION 371 10 5 ˇx 4
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5.4. EDM DEFINITION 373 corrected b
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5.4. EDM DEFINITION 375 by translat
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5.4. EDM DEFINITION 377 Crippen & H
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5.4. EDM DEFINITION 379 where ([√
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5.4. EDM DEFINITION 381 because (A.
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5.5. INVARIANCE 383 5.5.1.0.1 Examp
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5.5. INVARIANCE 385 x 2 x 2 x 3 x 1
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5.6. INJECTIVITY OF D & UNIQUE RECO
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5.7. EMBEDDING IN AFFINE HULL 393 5
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5.7. EMBEDDING IN AFFINE HULL 395 F
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5.7. EMBEDDING IN AFFINE HULL 397 5
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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 413 of
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5.10. EDM-ENTRY COMPOSITION 415 The
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5.11. EDM INDEFINITENESS 417 5.11.1
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5.11. EDM INDEFINITENESS 419 we hav
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5.11. EDM INDEFINITENESS 421 So bec
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5.11. EDM INDEFINITENESS 423 where
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5.12. LIST RECONSTRUCTION 425 where
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5.12. LIST RECONSTRUCTION 427 (a) (
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5.13. RECONSTRUCTION EXAMPLES 429 D
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5.13. RECONSTRUCTION EXAMPLES 431 T
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5.13. RECONSTRUCTION EXAMPLES 433 w
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5.14. FIFTH PROPERTY OF EUCLIDEAN M
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Chapter 6 Cone of distance matrices
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6.1. DEFINING EDM CONE 447 6.1 Defi
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6.2. POLYHEDRAL BOUNDS 449 This con
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6.3. √ EDM CONE IS NOT CONVEX 451
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6.4. A GEOMETRY OF COMPLETION 453 (
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6.4. A GEOMETRY OF COMPLETION 455 (
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6.4. A GEOMETRY OF COMPLETION 457 F
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6.5. EDM DEFINITION IN 11 T 459 by
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6.5. EDM DEFINITION IN 11 T 461 6.5
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6.5. EDM DEFINITION IN 11 T 463 1 0
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6.5. EDM DEFINITION IN 11 T 465 6.5
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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 477 When the Fin
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6.8. DUAL EDM CONE 479 Proof. First
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6.8. DUAL EDM CONE 481 EDM 2 = S 2
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6.8. DUAL EDM CONE 483 whose veraci
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6.8. DUAL EDM CONE 485 6.8.1.3.1 Ex
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6.8. DUAL EDM CONE 487 has dual aff
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6.8. DUAL EDM CONE 489 6.8.1.7 Scho
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6.9. THEOREM OF THE ALTERNATIVE 491
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6.10. POSTSCRIPT 493 When D is an E
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Chapter 7 Proximity problems In sum
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497 project on the subspace, then p
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499 H S N h 0 EDM N K = S N h ∩ R
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501 7.0.3 Problem approach Problems
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7.1. FIRST PREVALENT PROBLEM: 503 f
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7.1. FIRST PREVALENT PROBLEM: 505 7
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7.1. FIRST PREVALENT PROBLEM: 507 d
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7.1. FIRST PREVALENT PROBLEM: 509 7
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7.1. FIRST PREVALENT PROBLEM: 511 w
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7.1. FIRST PREVALENT PROBLEM: 513 T
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7.2. SECOND PREVALENT PROBLEM: 515
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7.3. THIRD PREVALENT PROBLEM: 525 g
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7.3. THIRD PREVALENT PROBLEM: 527 w
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7.3. THIRD PREVALENT PROBLEM: 529 7
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7.3. THIRD PREVALENT PROBLEM: 531 7
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7.3. THIRD PREVALENT PROBLEM: 533 O
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7.4. CONCLUSION 535 The rank constr
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Appendix A Linear algebra A.1 Main-
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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 545 (AB) T
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A.3. PROPER STATEMENTS 547 A.3.1 Se
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A.3. PROPER STATEMENTS 549 For A di
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A.3. PROPER STATEMENTS 551 Diagonal
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A.3. PROPER STATEMENTS 553 For A,B
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A.3. PROPER STATEMENTS 555 A.3.1.0.
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A.4. SCHUR COMPLEMENT 557 A.4 Schur
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A.4. SCHUR COMPLEMENT 559 A.4.0.0.2
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A.5. EIGEN DECOMPOSITION 561 When B
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A.5. EIGEN DECOMPOSITION 563 dim N(
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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 569 Given symmetric matr
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A.7. ZEROS 571 (TRANSPOSE.) Likewis
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A.7. ZEROS 573 For X,A∈ S M + [31
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A.7. ZEROS 575 A.7.5.0.1 Propositio
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Appendix B Simple matrices Mathemat
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B.1. RANK-ONE MATRIX (DYAD) 579 R(v
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B.1. RANK-ONE MATRIX (DYAD) 581 ran
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B.2. DOUBLET 583 R([u v ]) R(Π)= R
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B.3. ELEMENTARY MATRIX 585 If λ
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B.4. AUXILIARY V -MATRICES 587 the
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B.4. AUXILIARY V -MATRICES 589 18.
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B.5. ORTHOGONAL MATRIX 591 B.5 Orth
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B.5. ORTHOGONAL MATRIX 593 Figure 1
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Appendix C Some analytical optimal
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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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C.4. TWO-SIDED ORTHOGONAL PROCRUSTE
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Appendix D Matrix calculus From too
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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 E Projection For any A∈
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641 U T = U † for orthonormal (in
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E.1. IDEMPOTENT MATRICES 643 where
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E.1. IDEMPOTENT MATRICES 645 order,
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E.1. IDEMPOTENT MATRICES 647 When t
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E.3. SYMMETRIC IDEMPOTENT MATRICES
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E.5. PROJECTION EXAMPLES 655 E.4.0.
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E.5. PROJECTION EXAMPLES 657 a ∗
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E.5. PROJECTION EXAMPLES 659 E.5.0.
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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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Page 705 and 706: Appendix F Notation and a few defin
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Page 717 and 718: 717 O O sort-index matrix order of
Page 719 and 720: (x,y) angle between vectors x and y
Page 721 and 722: Bibliography [1] Suliman Al-Homidan
Page 723: BIBLIOGRAPHY 723 [24] Alexander I.
Page 727 and 728: BIBLIOGRAPHY 727 [78] Frank Critchl
Page 729 and 730: BIBLIOGRAPHY 729 [105] Richard L. D
Page 731 and 732: BIBLIOGRAPHY 731 [132] Michel X. Go
Page 733 and 734: BIBLIOGRAPHY 733 [162] T. Herrmann,
Page 735 and 736: BIBLIOGRAPHY 735 [191] Mark Kahrs a
Page 737 and 738: BIBLIOGRAPHY 737 [220] K. V. Mardia
Page 739 and 740: BIBLIOGRAPHY 739 [250] Pythagoras P
Page 741 and 742: BIBLIOGRAPHY 741 [277] Anthony Man-
Page 743 and 744: BIBLIOGRAPHY 743 [306] Michael W. T
Page 745 and 746: [333] Margaret H. Wright. The inter
Page 747 and 748: Index 0-norm, 203, 261, 294, 296, 2
Page 749 and 750: INDEX 749 product, 43, 92, 147, 254
Page 751 and 752: INDEX 751 coordinates, 140, 170, 17
Page 753 and 754: INDEX 753 affine dimension, 485 fea
Page 755 and 756: INDEX 755 affine, 209 nonlinear, 19
Page 757 and 758: INDEX 757 of point, 37 ray, 90 rela
Page 759 and 760: INDEX 759 normal, 47, 548, 563 norm
Page 761 and 762: INDEX 761 strictly, 515, 520 functi
Page 763 and 764: INDEX 763 vector, 45, 241, 248, 325
Page 765 and 766: INDEX 765 convex envelope, see conv
Page 767 and 768: INDEX 767 cone, 418, 420, 507 dual,
Page 769 and 770: INDEX 769 trilateration, 21, 42, 36
Page 772: Convex Optimization & Euclidean Dis
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