v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
760 APPENDIX E. PROJECTION10 0 idist(P 2 X i , S n +)10 −10 2 4 6 8 10 12 14 16 18Figure 172: Distance (confer (2077)) between positive semidefinite cone anditerate (2080) in affine subset A (2073) for Laurent’s completion problem;initially, decreasing geometrically.Using this technique, we find a positive semidefinite completion for⎡⎢⎣4 3 ? 23 4 3 ?? 3 4 32 ? 3 4⎤⎥⎦ (2081)Initializing the unknown entries to 0, they all converge geometrically to1.5858 (rounded) after about 42 iterations.Laurent gives a problem for which no positive semidefinite completionexists: [236]⎡⎢⎣1 1 ? 01 1 1 ?? 1 1 10 ? 1 1⎤⎥⎦ (2082)Initializing unknowns to 0, by alternating projection we find the constrained
E.10. ALTERNATING PROJECTION 761matrix closest to the positive semidefinite cone,⎡⎢⎣1 1 0.5454 01 1 1 0.54540.5454 1 1 10 0.5454 1 1⎤⎥⎦ (2083)and we find the positive semidefinite matrix closest to the affine subset A(2073):⎡⎤1.0521 0.9409 0.5454 0.0292⎢ 0.9409 1.0980 0.9451 0.5454⎥⎣ 0.5454 0.9451 1.0980 0.9409 ⎦ (2084)0.0292 0.5454 0.9409 1.0521These matrices (2083) and (2084) attain the Euclidean distance dist(A , S+).nConvergence is illustrated in Figure 172.E.10.3Optimization and projectionUnique projection on the nonempty intersection of arbitrary convex sets tofind the closest point therein is a convex optimization problem. The firstsuccessful application of alternating projection to this problem is attributedto Dykstra [127] [62] who in 1983 provided an elegant algorithm that prevailstoday. In 1988, Han [178] rediscovered the algorithm and provided aprimal−dual convergence proof. A synopsis of the history of alternatingprojection E.21 can be found in [64] where it becomes apparent that Dykstra’swork is seminal.E.10.3.1Dykstra’s algorithmAssume we are given some point b ∈ R n and closed convex sets{C k ⊂ R n | k=1... L}. Let x ki ∈ R n and y ki ∈ R n respectively denote aprimal and dual vector (whose meaning can be deduced from Figure 173and Figure 174) associated with set k at iteration i . Initializey k0 = 0 ∀k=1... L and x 1,0 = b (2085)E.21 For a synopsis of alternating projection applied to distance geometry, see [353,3.1].
- Page 709 and 710: E.3. SYMMETRIC IDEMPOTENT MATRICES
- Page 711 and 712: E.3. SYMMETRIC IDEMPOTENT MATRICES
- Page 713 and 714: E.3. SYMMETRIC IDEMPOTENT MATRICES
- Page 715 and 716: E.4. ALGEBRA OF PROJECTION ON AFFIN
- Page 717 and 718: E.5. PROJECTION EXAMPLES 717a ∗ 2
- Page 719 and 720: E.5. PROJECTION EXAMPLES 719where Y
- Page 721 and 722: E.5. PROJECTION EXAMPLES 721(B.4.2)
- Page 723 and 724: E.6. VECTORIZATION INTERPRETATION,
- Page 725 and 726: E.6. VECTORIZATION INTERPRETATION,
- Page 727 and 728: E.6. VECTORIZATION INTERPRETATION,
- Page 729 and 730: E.7. PROJECTION ON MATRIX SUBSPACES
- Page 731 and 732: E.7. PROJECTION ON MATRIX SUBSPACES
- Page 733 and 734: E.8. RANGE/ROWSPACE INTERPRETATION
- Page 735 and 736: E.9. PROJECTION ON CONVEX SET 735As
- Page 737 and 738: E.9. PROJECTION ON CONVEX SET 737Wi
- Page 739 and 740: E.9. PROJECTION ON CONVEX SET 739R(
- Page 741 and 742: E.9. PROJECTION ON CONVEX SET 741E.
- Page 743 and 744: E.9. PROJECTION ON CONVEX SET 743E.
- Page 745 and 746: E.9. PROJECTION ON CONVEX SET 745Un
- Page 747 and 748: E.9. PROJECTION ON CONVEX SET 747ac
- Page 749 and 750: E.10. ALTERNATING PROJECTION 749bC
- Page 751 and 752: E.10. ALTERNATING PROJECTION 7510
- Page 753 and 754: E.10. ALTERNATING PROJECTION 753E.1
- Page 755 and 756: E.10. ALTERNATING PROJECTION 755y 2
- Page 757 and 758: E.10. ALTERNATING PROJECTION 757Def
- Page 759: E.10. ALTERNATING PROJECTION 759Dis
- Page 763 and 764: E.10. ALTERNATING PROJECTION 763K
- Page 765 and 766: E.10. ALTERNATING PROJECTION 765E.1
- Page 767 and 768: E.10. ALTERNATING PROJECTION 767E.1
- Page 769 and 770: Appendix FNotation and a few defini
- Page 771 and 772: 771A ij or A(i, j) , ij th entry of
- Page 773 and 774: 773⊞orthogonal vector sum of sets
- Page 775 and 776: 775x +vector x whose negative entri
- Page 777 and 778: 777X point list ((76) having cardin
- Page 779 and 780: 779SDPSVDSNRdBEDMS n 1S n hS n⊥hS
- Page 781 and 782: 781vectorentrycubixquartixfeasible
- Page 783 and 784: 783Oorder of magnitude information
- Page 785 and 786: 785cofmatrix of cofactors correspon
- Page 787 and 788: Bibliography[1] Edwin A. Abbott. Fl
- Page 789 and 790: BIBLIOGRAPHY 789[23] Dror Baron, Mi
- Page 791 and 792: BIBLIOGRAPHY 791[49] Leonard M. Blu
- Page 793 and 794: BIBLIOGRAPHY 793[74] Yves Chabrilla
- Page 795 and 796: BIBLIOGRAPHY 795[102] Etienne de Kl
- Page 797 and 798: BIBLIOGRAPHY 797[129] Carl Eckart a
- Page 799 and 800: BIBLIOGRAPHY 799[154] James Gleik.
- Page 801 and 802: BIBLIOGRAPHY 801[182] Johan Håstad
- Page 803 and 804: BIBLIOGRAPHY 803[212] Viren Jain an
- Page 805 and 806: BIBLIOGRAPHY 805[237] Monique Laure
- Page 807 and 808: BIBLIOGRAPHY 807[265] Sunderarajan
- Page 809 and 810: BIBLIOGRAPHY 809Notes in Computer S
760 APPENDIX E. PROJECTION10 0 idist(P 2 X i , S n +)10 −10 2 4 6 8 10 12 14 16 18Figure 172: Distance (confer (2077)) between positive semidefinite cone anditerate (2080) in affine subset A (2073) for Laurent’s completion problem;initially, decreasing geometrically.Using this technique, we find a positive semidefinite completion for⎡⎢⎣4 3 ? 23 4 3 ?? 3 4 32 ? 3 4⎤⎥⎦ (2081)Initializing the unknown entries to 0, they all converge geometrically to1.5858 (rounded) after about 42 iterations.Laurent gives a problem for which no positive semidefinite completionexists: [236]⎡⎢⎣1 1 ? 01 1 1 ?? 1 1 10 ? 1 1⎤⎥⎦ (2082)Initializing unknowns to 0, by alternating projection we find the constrained
Change language