v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
548 CHAPTER 7. PROXIMITY PROBLEMS7.0.1 Measurement matrix HIdeally, we want a given matrix of measurements H ∈ R N×N to conformwith the first three Euclidean metric properties (5.2); to belong to theintersection of the orthant of nonnegative matrices R N×N+ with the symmetrichollow subspace S N h (2.2.3.0.1). Geometrically, we want H to belong to thepolyhedral cone (2.12.1.0.1)K S N h ∩ R N×N+ (1301)Yet in practice, H can possess significant measurement uncertainty (noise).Sometimes realization of an optimization problem demands that itsinput, the given matrix H , possess some particular characteristics; perhapssymmetry and hollowness or nonnegativity. When that H given does nothave the desired properties, then we must impose them upon H prior tooptimization:When measurement matrix H is not symmetric or hollow, taking itssymmetric hollow part is equivalent to orthogonal projection on thesymmetric hollow subspace S N h .When measurements of distance in H are negative, zeroing negativeentries effects unique minimum-distance projection on the orthant ofnonnegative matrices R N×N+ in isomorphic R N2 (E.9.2.2.3).7.0.1.1 Order of impositionSince convex cone K (1301) is the intersection of an orthant with a subspace,we want to project on that subset of the orthant belonging to the subspace;on the nonnegative orthant in the symmetric hollow subspace that is, in fact,the intersection. For that reason alone, unique minimum-distance projectionof H on K (that member of K closest to H in isomorphic R N2 in the Euclideansense) can be attained by first taking its symmetric hollow part, and onlythen clipping negative entries of the result to 0 ; id est, there is only onecorrect order of projection, in general, on an orthant intersecting a subspace:
549project on the subspace, then project the result on the orthant in thatsubspace. (conferE.9.5)In contrast, order of projection on an intersection of subspaces is arbitrary.That order-of-projection rule applies more generally, of course, tothe intersection of any convex set C with any subspace. Consider theproximity problem 7.1 over convex feasible set S N h ∩ C given nonsymmetricnonhollow H ∈ R N×N :minimizeB∈S N hsubject to‖B − H‖ 2 FB ∈ C(1302)a convex optimization problem. Because the symmetric hollow subspace S N his orthogonal to the antisymmetric antihollow subspace R N×N⊥h(2.2.3), thenfor B ∈ S N h( ( ))1tr B T 2 (H −HT ) + δ 2 (H) = 0 (1303)so the objective function is equivalent to( )∥ ‖B − H‖ 2 F ≡1 ∥∥∥2∥ B − 2 (H +HT ) − δ 2 (H) +F21∥2 (H −HT ) + δ 2 (H)∥F(1304)This means the antisymmetric antihollow part of given matrix H wouldbe ignored by minimization with respect to symmetric hollow variable Bunder Frobenius’ norm; id est, minimization proceeds as though given thesymmetric hollow part of H .This action of Frobenius’ norm (1304) is effectively a Euclidean projection(minimum-distance projection) of H on the symmetric hollow subspace S N hprior to minimization. Thus minimization proceeds inherently following thecorrect order for projection on S N h ∩ C . Therefore we may either assumeH ∈ S N h , or take its symmetric hollow part prior to optimization.7.1 There are two equivalent interpretations of projection (E.9): one finds a set normal,the other, minimum distance between a point and a set. Here we realize the latter view.
- Page 497 and 498: Chapter 6Cone of distance matricesF
- 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: Chapter 7Proximity problemsIn the
- Page 551 and 552: 551HS N h0EDM NK = S N h ∩ R N×N
- 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
548 CHAPTER 7. PROXIMITY PROBLEMS7.0.1 Measurement matrix HIdeally, we want a given matrix of measurements H ∈ R N×N to conformwith the first three Euclidean metric properties (5.2); to belong to theintersection of the orthant of nonnegative matrices R N×N+ with the symmetrichollow subspace S N h (2.2.3.0.1). Geometrically, we want H to belong to thepolyhedral cone (2.12.1.0.1)K S N h ∩ R N×N+ (1301)Yet in practice, H can possess significant measurement uncertainty (noise).Sometimes realization of an optimization problem demands that itsinput, the given matrix H , possess some particular characteristics; perhapssymmetry and hollowness or nonnegativity. When that H given does nothave the desired properties, then we must impose them upon H prior tooptimization:When measurement matrix H is not symmetric or hollow, taking itssymmetric hollow part is equivalent to orthogonal projection on thesymmetric hollow subspace S N h .When measurements of distance in H are negative, zeroing negativeentries effects unique minimum-distance projection on the orthant ofnonnegative matrices R N×N+ in isomorphic R N2 (E.9.2.2.3).7.0.1.1 Order of impositionSince convex cone K (1301) is the intersection of an orthant with a subspace,we want to project on that subset of the orthant belonging to the subspace;on the nonnegative orthant in the symmetric hollow subspace that is, in fact,the intersection. For that reason alone, unique minimum-distance projectionof H on K (that member of K closest to H in isomorphic R N2 in the Euclideansense) can be attained by first taking its symmetric hollow part, and onlythen clipping negative entries of the result to 0 ; id est, there is only onecorrect order of projection, in general, on an orthant intersecting a subspace: