v2008.02.02 - Convex Optimization
v2008.02.02 - Convex Optimization v2008.02.02 - Convex Optimization
456 CHAPTER 7. PROXIMITY PROBLEMS......❜..❜..❜..❜..❜..❜..❜.❜❜❜❜❜❜❜✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ 0 ✟✟✟✟✟✟✟✟✟✟✟EDM N❜❜ S N ❝ ❝❝❝❝❝❝❝❝❝❝❝h❜❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❜❜❜❜❜❜ K = S N .h ∩ R N×N+.❜... S N ❜.. ❜..❜.. ❜❜❜ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧....... . . . ..R N×N........................Figure 114: Pseudo-Venn diagram: The EDM cone belongs to theintersection of the symmetric hollow subspace with the nonnegative orthant;EDM N ⊆ K (733). EDM N cannot exist outside S N h , but R N×N+ does.......7.0.1.2 Egregious input error under nonnegativity demandMore pertinent to the optimization problems presented herein whereC ∆ = EDM N ⊆ K = S N h ∩ R N×N+ (1137)then should some particular realization of a proximity problem demandinput H be nonnegative, and were we only to zero negative entries of anonsymmetric nonhollow input H prior to optimization, then the ensuingprojection on EDM N would be guaranteed incorrect (out of order).Now comes a surprising fact: Even were we to correctly follow theorder-of-projection rule and provide H ∈ K prior to optimization, then theensuing projection on EDM N will be incorrect whenever input H has negativeentries and some proximity problem demands nonnegative input H .
457HS N h0EDM NK = S N h ∩ R N×N+Figure 115: Pseudo-Venn diagram from Figure 114 showing elbow placedin path of projection of H on EDM N ⊂ S N h by an optimization problemdemanding nonnegative input matrix H . The first two line segmentsleading away from H result from correct order-of-projection required toprovide nonnegative H prior to optimization. Were H nonnegative, then itsprojection on S N h would instead belong to K ; making the elbow disappear.(confer Figure 126)
- Page 405 and 406: 6.1. DEFINING EDM CONE 4056.1 Defin
- Page 407 and 408: 6.2. POLYHEDRAL BOUNDS 407This cone
- Page 409 and 410: 6.3.√EDM CONE IS NOT CONVEX 409N
- Page 411 and 412: 6.4. A GEOMETRY OF COMPLETION 4116.
- Page 413 and 414: 6.4. A GEOMETRY OF COMPLETION 413(a
- Page 415 and 416: 6.4. A GEOMETRY OF COMPLETION 415Fi
- Page 417 and 418: 6.5. EDM DEFINITION IN 11 T 417and
- Page 419 and 420: 6.5. EDM DEFINITION IN 11 T 419then
- Page 421 and 422: 6.5. EDM DEFINITION IN 11 T 4216.5.
- Page 423 and 424: 6.5. EDM DEFINITION IN 11 T 423D =
- Page 425 and 426: 6.6. CORRESPONDENCE TO PSD CONE S N
- Page 427 and 428: 6.6. CORRESPONDENCE TO PSD CONE S N
- Page 429 and 430: 6.6. CORRESPONDENCE TO PSD CONE S N
- Page 431 and 432: 6.7. VECTORIZATION & PROJECTION INT
- Page 433 and 434: 6.7. VECTORIZATION & PROJECTION INT
- Page 435 and 436: 6.8. DUAL EDM CONE 435When the Fins
- Page 437 and 438: 6.8. DUAL EDM CONE 437Proof. First,
- Page 439 and 440: 6.8. DUAL EDM CONE 439EDM 2 = S 2 h
- Page 441 and 442: 6.8. DUAL EDM CONE 441whose veracit
- Page 443 and 444: 6.8. DUAL EDM CONE 4436.8.1.3.1 Exe
- Page 445 and 446: 6.8. DUAL EDM CONE 445has dual affi
- Page 447 and 448: 6.8. DUAL EDM CONE 4476.8.1.7 Schoe
- Page 449 and 450: 6.9. THEOREM OF THE ALTERNATIVE 449
- Page 451 and 452: 6.10. POSTSCRIPT 451When D is an ED
- Page 453 and 454: Chapter 7Proximity problemsIn summa
- Page 455: 455project on the subspace, then pr
- Page 459 and 460: 4597.0.3 Problem approachProblems t
- Page 461 and 462: 7.1. FIRST PREVALENT PROBLEM: 461fi
- Page 463 and 464: 7.1. FIRST PREVALENT PROBLEM: 4637.
- Page 465 and 466: 7.1. FIRST PREVALENT PROBLEM: 465di
- Page 467 and 468: 7.1. FIRST PREVALENT PROBLEM: 4677.
- Page 469 and 470: 7.1. FIRST PREVALENT PROBLEM: 469wh
- Page 471 and 472: 7.1. FIRST PREVALENT PROBLEM: 471Th
- Page 473 and 474: 7.2. SECOND PREVALENT PROBLEM: 473O
- Page 475 and 476: 7.2. SECOND PREVALENT PROBLEM: 475S
- Page 477 and 478: 7.2. SECOND PREVALENT PROBLEM: 477r
- Page 479 and 480: 7.2. SECOND PREVALENT PROBLEM: 479c
- Page 481 and 482: 7.2. SECOND PREVALENT PROBLEM: 4817
- Page 483 and 484: 7.3. THIRD PREVALENT PROBLEM: 483gl
- Page 485 and 486: 7.3. THIRD PREVALENT PROBLEM: 485wh
- Page 487 and 488: 7.3. THIRD PREVALENT PROBLEM: 4877.
- Page 489 and 490: 7.3. THIRD PREVALENT PROBLEM: 4897.
- Page 491 and 492: 7.3. THIRD PREVALENT PROBLEM: 491Ou
- Page 493 and 494: 7.4. CONCLUSION 493The rank constra
- Page 495 and 496: Appendix ALinear algebraA.1 Main-di
- Page 497 and 498: A.1. MAIN-DIAGONAL δ OPERATOR, λ
- Page 499 and 500: A.2. SEMIDEFINITENESS: DOMAIN OF TE
- Page 501 and 502: A.2. SEMIDEFINITENESS: DOMAIN OF TE
- Page 503 and 504: A.3. PROPER STATEMENTS 503A.3.0.0.1
- Page 505 and 506: A.3. PROPER STATEMENTS 505By simila
456 CHAPTER 7. PROXIMITY PROBLEMS......❜..❜..❜..❜..❜..❜..❜.❜❜❜❜❜❜❜✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ 0 ✟✟✟✟✟✟✟✟✟✟✟EDM N❜❜ S N ❝ ❝❝❝❝❝❝❝❝❝❝❝h❜❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❜❜❜❜❜❜ K = S N .h ∩ R N×N+.❜... S N ❜.. ❜..❜.. ❜❜❜ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧....... . . . ..R N×N........................Figure 114: Pseudo-Venn diagram: The EDM cone belongs to theintersection of the symmetric hollow subspace with the nonnegative orthant;EDM N ⊆ K (733). EDM N cannot exist outside S N h , but R N×N+ does.......7.0.1.2 Egregious input error under nonnegativity demandMore pertinent to the optimization problems presented herein whereC ∆ = EDM N ⊆ K = S N h ∩ R N×N+ (1137)then should some particular realization of a proximity problem demandinput H be nonnegative, and were we only to zero negative entries of anonsymmetric nonhollow input H prior to optimization, then the ensuingprojection on EDM N would be guaranteed incorrect (out of order).Now comes a surprising fact: Even were we to correctly follow theorder-of-projection rule and provide H ∈ K prior to optimization, then theensuing projection on EDM N will be incorrect whenever input H has negativeentries and some proximity problem demands nonnegative input H .