v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
550 CHAPTER 7. PROXIMITY PROBLEMS......❜..❜..❜..❜..❜..❜..❜.❜❜❜❜❜❜❜✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧ 0EDM N❜❜ S N ❝ ✟✟✟✟✟✟✟✟✟✟✟❝❝❝❝❝❝❝❝❝❝❝h❜❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❜❜❜❜❜❜ K = S N .h ∩ R N×N+.❜... S N ❜.. ❜..❜.. ❜❜❜ ✧ ✧✧✧✧✧✧✧✧✧✧✧✧✧✧....... . . . ..R N×N........................Figure 152: Pseudo-Venn diagram: The EDM cone belongs to theintersection of the symmetric hollow subspace with the nonnegative orthant;EDM N ⊆ K (890). 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+ (1305)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 .
551HS N h0EDM NK = S N h ∩ R N×N+Figure 153: Pseudo-Venn diagram from Figure 152 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 166)
- Page 499 and 500: 6.1. DEFINING EDM CONE 4996.1 Defin
- Page 501 and 502: 6.2. POLYHEDRAL BOUNDS 501This cone
- Page 503 and 504: 6.4. EDM DEFINITION IN 11 T 503That
- Page 505 and 506: 6.4. EDM DEFINITION IN 11 T 505N(1
- Page 507 and 508: 6.4. EDM DEFINITION IN 11 T 507Then
- 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
- Page 521 and 522: 6.6. VECTORIZATION & PROJECTION INT
- Page 523 and 524: 6.7. A GEOMETRY OF COMPLETION 523(b
- Page 525 and 526: 6.7. A GEOMETRY OF COMPLETION 525[3
- Page 527 and 528: 6.7. A GEOMETRY OF COMPLETION 527wh
- Page 529 and 530: 6.8. DUAL EDM CONE 529to the geomet
- Page 531 and 532: 6.8. DUAL EDM CONE 531Proof. First,
- Page 533 and 534: 6.8. DUAL EDM CONE 533EDM 2 = S 2 h
- Page 535 and 536: 6.8. DUAL EDM CONE 535therefore the
- Page 537 and 538: 6.8. DUAL EDM CONE 537Elegance of t
- Page 539 and 540: 6.8. DUAL EDM CONE 5396.8.1.5 Affin
- Page 541 and 542: 6.8. DUAL EDM CONE 5416.8.1.7 Schoe
- Page 543 and 544: 6.8. DUAL EDM CONE 5430dvec rel ∂
- Page 545 and 546: 6.10. POSTSCRIPT 5456.10 Postscript
- Page 547 and 548: Chapter 7Proximity problemsIn the
- Page 549: 549project on the subspace, then pr
- Page 553 and 554: 5537.0.3 Problem approachProblems t
- Page 555 and 556: 7.1. FIRST PREVALENT PROBLEM: 555fi
- Page 557 and 558: 7.1. FIRST PREVALENT PROBLEM: 5577.
- Page 559 and 560: 7.1. FIRST PREVALENT PROBLEM: 559di
- Page 561 and 562: 7.1. FIRST PREVALENT PROBLEM: 5617.
- Page 563 and 564: 7.1. FIRST PREVALENT PROBLEM: 563wh
- Page 565 and 566: 7.1. FIRST PREVALENT PROBLEM: 565Th
- Page 567 and 568: 7.2. SECOND PREVALENT PROBLEM: 567O
- Page 569 and 570: 7.2. SECOND PREVALENT PROBLEM: 569S
- Page 571 and 572: 7.2. SECOND PREVALENT PROBLEM: 571r
- Page 573 and 574: 7.2. SECOND PREVALENT PROBLEM: 573w
- Page 575 and 576: 7.2. SECOND PREVALENT PROBLEM: 5757
- Page 577 and 578: 7.2. SECOND PREVALENT PROBLEM: 577a
- Page 579 and 580: 7.3. THIRD PREVALENT PROBLEM: 579is
- Page 581 and 582: 7.3. THIRD PREVALENT PROBLEM: 581We
- Page 583 and 584: 7.3. THIRD PREVALENT PROBLEM: 583su
- Page 585 and 586: 7.3. THIRD PREVALENT PROBLEM: 585Gi
- Page 587 and 588: 7.3. THIRD PREVALENT PROBLEM: 587Op
- Page 589 and 590: 7.4. CONCLUSION 589filtering, multi
- Page 591 and 592: Appendix ALinear algebraA.1 Main-di
- Page 593 and 594: A.1. MAIN-DIAGONAL δ OPERATOR, λ
- Page 595 and 596: A.1. MAIN-DIAGONAL δ OPERATOR, λ
- Page 597 and 598: A.2. SEMIDEFINITENESS: DOMAIN OF TE
- Page 599 and 600: A.3. PROPER STATEMENTS 599(AB) T
551HS N h0EDM NK = S N h ∩ R N×N+Figure 153: Pseudo-Venn diagram from Figure 152 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 166)
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