v2009.01.01 - Convex Optimization

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296 CHAPTER 4. SEMIDEFINITE PROGRAMMING The set {u∈ R n | cardu = n−k , u i ∈ {0, 1}} comprises the extreme points of set (684) which is a nonnegative hypercube slice. An optimal solution y to (467), that is an extreme point of its feasible set, is known in closed form: it has 1 in each entry corresponding to the n−k smallest entries of x ⋆ and has 0 elsewhere. By Proposition 7.1.3.0.3, the polar of that particular direction −y can be interpreted 4.30 as pointing toward the set of all vectors having the same ordering as x ⋆ and cardinality-k . When y = 1, as in 1-norm minimization for example, then polar direction −y points directly at the origin (the cardinality-0 vector). 4.5.1.2 convergence <strong>Convex</strong> iteration (143) (467) always converges to a locally optimal solution by virtue of a monotonically nonincreasing real objective sequence. [222,1.2] [37,1.1] There can be no proof of global optimality, defined by (683). Constraining cardinality, solution to problem (682), can often be achieved but simple examples can be contrived that cause an iteration to stall at a solution of undesirable cardinality. The direction vector is then manipulated to steer out of local minima: 4.5.1.2.1 Example. Sparsest solution to Ax = b. [61] [100] Problem (616) has sparsest solution not easily recoverable by least 1-norm; id est, not by compressed sensing because of proximity to a theoretical lower bound on number of measurements m depicted in Figure 80: for A∈ R m×n Given data from Example 4.2.3.1.1, for m=3, n=6, k=1 ⎡ ⎤ ⎡ ⎤ −1 1 8 1 1 0 1 ⎢ 1 1 1 A = ⎣ −3 2 8 − 1 ⎥ ⎢ 2 3 2 3 ⎦ , b = ⎣ −9 4 8 1 4 1 9 1 4 − 1 9 1 2 1 4 ⎥ ⎦ (616) the sparsest solution to classical linear equation Ax = b is x = e 4 ∈ R 6 (confer (629)). Although the sparsest solution is recoverable by inspection, we discern it instead by convex iteration; namely, by iterating problem sequence 4.30 <strong>Convex</strong> iteration (143) (467) is not a projection method because there is no thresholding or discard of variable-vector x entries.
4.5. CONSTRAINING CARDINALITY 297 m/k 7 6 Donoho bound approximation x > 0 constraint 5 4 3 m > k log 2 (1+n/k) minimize ‖x‖ 1 x subject to Ax = b 2 1 0 0.2 0.4 0.6 0.8 1 Figure 80: [22] Donoho least lower bound on number of measurements m below which recovery of k-sparse n-length signal x by linear programming fails with overwhelming probability. Hard problems are below curve, but not the reverse; id est, failure above depends on proximity. Inequality demarcates approximation. Problems having nonnegativity constraint are easier to solve. k/n
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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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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
Page 245 and 246: Chapter 4 Semidefinite programming
Page 247 and 248: 4.1. CONIC PROBLEM 247 where K is a
Page 249 and 250: 4.1. CONIC PROBLEM 249 4.1.1.2 Redu
Page 251 and 252: 4.1. CONIC PROBLEM 251 In any SDP f
Page 253 and 254: 4.1. CONIC PROBLEM 253 Proposition
Page 255 and 256: 4.2. FRAMEWORK 255 sets are closed
Page 257 and 258: 4.2. FRAMEWORK 257 4.2.1.1.3 Exampl
Page 259 and 260: 4.2. FRAMEWORK 259 4.2.2 Duals The
Page 261 and 262: 4.2. FRAMEWORK 261 When equality is
Page 263 and 264: 4.2. FRAMEWORK 263 The pseudoinvers
Page 265 and 266: 4.2. FRAMEWORK 265 For the data giv
Page 267 and 268: 4.2. FRAMEWORK 267 minimizes an aff
Page 269 and 270: 4.3. RANK REDUCTION 269 whose rank
Page 271 and 272: 4.3. RANK REDUCTION 271 and where m
Page 273 and 274: 4.3. RANK REDUCTION 273 4.3.3 Optim
Page 275 and 276: 4.3. RANK REDUCTION 275 Initialize:
Page 277 and 278: 4.4. RANK-CONSTRAINED SEMIDEFINITE
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Page 345 and 346: 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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E.8. RANGE/ROWSPACE INTERPRETATION
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E.9. PROJECTION ON CONVEX SET 675 A
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E.9. PROJECTION ON CONVEX SET 677 W
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E.9. PROJECTION ON CONVEX SET 679 P
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E.9. PROJECTION ON CONVEX SET 681 E
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E.9. PROJECTION ON CONVEX SET 683 T
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E.9. PROJECTION ON CONVEX SET 685
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E.10. ALTERNATING PROJECTION 687 E.
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E.10. ALTERNATING PROJECTION 689 b
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E.10. ALTERNATING PROJECTION 691 a
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E.10. ALTERNATING PROJECTION 693 (a
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E.10. ALTERNATING PROJECTION 695 wh
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E.10. ALTERNATING PROJECTION 699 10
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E.10. ALTERNATING PROJECTION 703 E
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Appendix F Notation and a few defin
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707 a.i. c.i. l.i. w.r.t affinely i
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709 is or ← → t → 0 + as in
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711 ∑ π(γ) Ξ Π ∏ ψ(Z) D D
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713 R m×n Euclidean vector space o
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715 H − H + ∂H ∂H ∂H −
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717 O O sort-index matrix order of
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(x,y) angle between vectors x and y
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Bibliography [1] Suliman Al-Homidan
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BIBLIOGRAPHY 723 [24] Alexander I.
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BIBLIOGRAPHY 725 [52] Stephen Boyd,
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BIBLIOGRAPHY 727 [78] Frank Critchl
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BIBLIOGRAPHY 729 [105] Richard L. D
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BIBLIOGRAPHY 731 [132] Michel X. Go
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BIBLIOGRAPHY 733 [162] T. Herrmann,
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BIBLIOGRAPHY 735 [191] Mark Kahrs a
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BIBLIOGRAPHY 737 [220] K. V. Mardia
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BIBLIOGRAPHY 739 [250] Pythagoras P
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BIBLIOGRAPHY 741 [277] Anthony Man-
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BIBLIOGRAPHY 743 [306] Michael W. T
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[333] Margaret H. Wright. The inter
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Index 0-norm, 203, 261, 294, 296, 2
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INDEX 749 product, 43, 92, 147, 254
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INDEX 751 coordinates, 140, 170, 17
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INDEX 753 affine dimension, 485 fea
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INDEX 755 affine, 209 nonlinear, 19
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INDEX 757 of point, 37 ray, 90 rela
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INDEX 759 normal, 47, 548, 563 norm
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INDEX 761 strictly, 515, 520 functi
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INDEX 763 vector, 45, 241, 248, 325
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INDEX 765 convex envelope, see conv
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INDEX 767 cone, 418, 420, 507 dual,
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INDEX 769 trilateration, 21, 42, 36
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