v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
586 CHAPTER 7. PROXIMITY PROBLEMSProof. We justify disappearance of the hollowness constraint inconvex optimization problem (1408a): From the arguments in7.1.3with regard ⎡ to⎤π the permutation operator, cone membership constraintδ(Υ) ∈⎣ Rρ+1 +0 ⎦∩ ∂H from (1408a) is equivalent toR −⎡δ(Υ) ∈ ⎣ Rρ+1 +0R −⎤⎦ ∩ ∂H ∩ K M (1409)where K M is the monotone cone (2.13.9.4.3). Membership of δ(Υ) to thepolyhedral cone of majorization (Theorem A.1.2.0.1)K ∗ λδ = ∂H ∩ K ∗ M+ (1426)where K ∗ M+ is the dual monotone nonnegative cone (2.13.9.4.2), is acondition (in absence of a hollowness [ constraint) ] that would insure existence0 1Tof a symmetric hollow matrix . Curiously, intersection of1 −D⎡ ⎤this feasible superset ⎣ Rρ+1 +0 ⎦∩ ∂H ∩ K M from (1409) with the cone ofmajorization K ∗ λδR −is a benign operation; id est,∂H ∩ K ∗ M+ ∩ K M = ∂H ∩ K M (1410)verifiable by observing conic dependencies (2.10.3) among the aggregate ofhalfspace-description normals. The cone membership constraint in (1408a)therefore inherently insures existence of a symmetric hollow matrix.
7.3. THIRD PREVALENT PROBLEM: 587Optimization (1408b) would be a Procrustes problem (C.4) were itnot for the hollowness constraint; it is, instead, a minimization over theintersection of the nonconvex manifold of orthogonal matrices with anothernonconvex set in variable R specified by the hollowness constraint.We solve problem (1408b) by a method introduced in4.6.0.0.2: DefineR = [r 1 · · · r N+1 ]∈ R N+1×N+1 and make the assignment⎡G = ⎢⎣r 1.r N+11⎤[ r1 T · · · rN+1 T 1] ⎥∈ S (N+1)2 +1⎦⎡⎤ ⎡R 11 · · · R 1,N+1 r 1.....= ⎢⎣ R1,N+1 T ⎥R N+1,N+1 r N+1 ⎦ ⎢⎣r1 T · · · rN+1 T 1(1411)⎤r 1 r1 T · · · r 1 rN+1 T r 1.....r N+1 r1 T r N+1 rN+1 T ⎥r N+1 ⎦r1 T · · · rN+1 T 1where R ij r i rj T ∈ R N+1×N+1 and Υ ⋆ ii ∈ R . Since R Υ ⋆ R T = N+1 ∑ΥiiR ⋆ ii thenproblem (1408b) is equivalently expressed:∥ ∥∥∥ N+12∑minimize Υ ⋆R ii ∈S , R ij , r iiiR ii − Λ∥i=1Fsubject to trR ii = 1, i=1... N+1trR ij ⎡= 0,⎤i
- 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 and 550: 549project on the subspace, then pr
- 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: 7.3. THIRD PREVALENT PROBLEM: 585Gi
- 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
- Page 601 and 602: A.3. PROPER STATEMENTS 601A.3.1Semi
- Page 603 and 604: A.3. PROPER STATEMENTS 603For A dia
- Page 605 and 606: A.3. PROPER STATEMENTS 605Diagonali
- Page 607 and 608: A.3. PROPER STATEMENTS 607For A,B
- Page 609 and 610: A.3. PROPER STATEMENTS 609When B is
- Page 611 and 612: A.4. SCHUR COMPLEMENT 611A.4 Schur
- Page 613 and 614: A.4. SCHUR COMPLEMENT 613A.4.0.0.3
- Page 615 and 616: A.4. SCHUR COMPLEMENT 615From Corol
- Page 617 and 618: A.5. EIGENVALUE DECOMPOSITION 617wh
- Page 619 and 620: A.5. EIGENVALUE DECOMPOSITION 619A.
- Page 621 and 622: A.6. SINGULAR VALUE DECOMPOSITION,
- Page 623 and 624: A.6. SINGULAR VALUE DECOMPOSITION,
- Page 625 and 626: A.7. ZEROS 625A.6.5SVD of symmetric
- Page 627 and 628: A.7. ZEROS 627(Transpose.)Likewise,
- Page 629 and 630: A.7. ZEROS 629For X,A∈ S M +[34,2
- Page 631 and 632: A.7. ZEROS 631A.7.5.0.1 Proposition
- Page 633 and 634: Appendix BSimple matricesMathematic
- Page 635 and 636: B.1. RANK-ONE MATRIX (DYAD) 635R(v)
7.3. THIRD PREVALENT PROBLEM: 587<strong>Optimization</strong> (1408b) would be a Procrustes problem (C.4) were itnot for the hollowness constraint; it is, instead, a minimization over theintersection of the nonconvex manifold of orthogonal matrices with anothernonconvex set in variable R specified by the hollowness constraint.We solve problem (1408b) by a method introduced in4.6.0.0.2: DefineR = [r 1 · · · r N+1 ]∈ R N+1×N+1 and make the assignment⎡G = ⎢⎣r 1.r N+11⎤[ r1 T · · · rN+1 T 1] ⎥∈ S (N+1)2 +1⎦⎡⎤ ⎡R 11 · · · R 1,N+1 r 1.....= ⎢⎣ R1,N+1 T ⎥R N+1,N+1 r N+1 ⎦ ⎢⎣r1 T · · · rN+1 T 1(1411)⎤r 1 r1 T · · · r 1 rN+1 T r 1.....r N+1 r1 T r N+1 rN+1 T ⎥r N+1 ⎦r1 T · · · rN+1 T 1where R ij r i rj T ∈ R N+1×N+1 and Υ ⋆ ii ∈ R . Since R Υ ⋆ R T = N+1 ∑ΥiiR ⋆ ii thenproblem (1408b) is equivalently expressed:∥ ∥∥∥ N+12∑minimize Υ ⋆R ii ∈S , R ij , r iiiR ii − Λ∥i=1Fsubject to trR ii = 1, i=1... N+1trR ij ⎡= 0,⎤i