v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
754 APPENDIX E. PROJECTIONIf the intersection of two closed convex sets C 1 ∩ C 2 is empty, then the iteratesconverge to a point of minimum distance, a fixed point of the projectionproduct. Otherwise, convergence is to some fixed point in their intersection (afeasible solution) whose existence is guaranteed by virtue of the fact that eachand every point in the convex intersection is in one-to-one correspondencewith fixed points of the nonexpansive projection product.Bauschke & Borwein [28,2] argue that any sequence monotonic in thesense of Fejér is convergent: E.20E.10.2.0.1 Definition. Fejér monotonicity. [270]Given closed convex set C ≠ ∅ , then a sequence x i ∈ R n , i=0, 1, 2..., ismonotonic in the sense of Fejér with respect to C iff‖x i+1 − c‖ ≤ ‖x i − c‖ for all i≥0 and each and every c ∈ C (2058)Given x 0 b , if we express each iteration of alternating projection byx i+1 = P 1 P 2 x i , i=0, 1, 2... (2059)and define any fixed point a =P 1 P 2 a , then sequence x i is Fejér monotonewith respect to fixed point a because‖P 1 P 2 x i − a‖ ≤ ‖x i − a‖ ∀i ≥ 0 (2060)by nonexpansivity. The nonincreasing sequence ‖P 1 P 2 x i − a‖ is boundedbelow hence convergent because any bounded monotonic sequence in Ris convergent; [258,1.2] [42,1.1] P 1 P 2 x i+1 = P 1 P 2 x i = x i+1 . Sequencex i therefore converges to some fixed point. If the intersection C 1 ∩ C 2is nonempty, convergence is to some point there by the distance theorem.Otherwise, x i converges to a point in C 1 of minimum distance to C 2 .E.10.2.0.2 Example. Hyperplane/orthant intersection.Find a feasible solution (2052) belonging to the nonempty intersection of twoconvex sets: given A∈ R m×n , β ∈ R(A)C 1 ∩ C 2 = R n + ∩ A = {y | y ≽ 0} ∩ {y | Ay = β} ⊂ R n (2061)E.20 Other authors prove convergence by different means; e.g., [175] [63].△
E.10. ALTERNATING PROJECTION 755y 2θC 1 = R 2 +bPb = ( ∞ ∏j=1 k=12∏P k )by 1C 2 = A = {y | [ 1 1 ]y = 1}Figure 170: From Example E.10.2.0.2 in R 2 , showing von Neumann-stylealternating projection to find feasible solution belonging to intersection ofnonnegative orthant with hyperplane A . Point Pb lies at intersection ofhyperplane with ordinate axis. In this particular example, feasible solutionfound is coincidentally optimal. Rate of convergence depends upon angle θ ;as it becomes more acute, convergence slows. [175,3]∥ x ∏i − ( ∞j=1 k=12∏P k )b∥201816141210864200 5 10 15 20 25 30 35 40 45iFigure 171: Example E.10.2.0.2 in R 1000 ; geometric convergence of iteratesin norm.
- Page 703 and 704: E.1. IDEMPOTENT MATRICES 703where A
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- Page 733 and 734: E.8. RANGE/ROWSPACE INTERPRETATION
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- 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
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- Page 781 and 782: 781vectorentrycubixquartixfeasible
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- 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
E.10. ALTERNATING PROJECTION 755y 2θC 1 = R 2 +bPb = ( ∞ ∏j=1 k=12∏P k )by 1C 2 = A = {y | [ 1 1 ]y = 1}Figure 170: From Example E.10.2.0.2 in R 2 , showing von Neumann-stylealternating projection to find feasible solution belonging to intersection ofnonnegative orthant with hyperplane A . Point Pb lies at intersection ofhyperplane with ordinate axis. In this particular example, feasible solutionfound is coincidentally optimal. Rate of convergence depends upon angle θ ;as it becomes more acute, convergence slows. [175,3]∥ x ∏i − ( ∞j=1 k=12∏P k )b∥201816141210864200 5 10 15 20 25 30 35 40 45iFigure 171: Example E.10.2.0.2 in R 1000 ; geometric convergence of iteratesin norm.
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