v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
698 APPENDIX E. PROJECTION Barvinok (2.9.3.0.1) shows that if a point feasible with (1948) exists, then there exists an X ∈ A ∩ S n + such that ⌊√ ⌋ 8m + 1 − 1 rankX ≤ (245) 2 E.10.2.1.2 Example. Semidefinite matrix completion. Continuing Example E.10.2.1.1: When m≤n(n + 1)/2 and the A j matrices are distinct members of the standard orthonormal basis {E lq ∈ S n } (52) { } el e T {A j ∈ S n l , l = q = 1... n , j =1... m} ⊆ {E lq } = √1 2 (e l e T q + e q e T l ), 1 ≤ l < q ≤ n and when the constants b j X = ∆ [X lq ]∈ S n {b j , j =1... m} ⊆ (1954) are set to constrained entries of variable { } Xlq √ , l = q = 1... n X lq 2 , 1 ≤ l < q ≤ n = {〈X,E lq 〉} (1955) then the equality constraints in (1948) fix individual entries of X ∈ S n . Thus the feasibility problem becomes a positive semidefinite matrix completion problem. Projection of iterate X i ∈ S n on A simplifies to (confer (1950)) P 2 svec X i = svec X i − A T (A svec X i − b) (1956) From this we can see that orthogonal projection is achieved simply by setting corresponding entries of P 2 X i to the known entries of X , while the remaining entries of P 2 X i are set to corresponding entries of the current iterate X i . Using this technique, we find a positive semidefinite completion for ⎡ ⎤ 4 3 ? 2 ⎢ 3 4 3 ? ⎥ ⎣ ? 3 4 3 ⎦ (1957) 2 ? 3 4 Initializing the unknown entries to 0, they all converge geometrically to 1.5858 (rounded) after about 42 iterations.
E.10. ALTERNATING PROJECTION 699 10 0 dist(P 2 X i , S n +) 10 −1 0 2 4 6 8 10 12 14 16 18 i Figure 147: Distance (confer (1953)) between PSD cone and iterate (1956) in affine subset A (1949) for Laurent’s completion problem; initially, decreasing geometrically. Laurent gives a problem for which no positive semidefinite completion exists: [203] ⎡ ⎤ 1 1 ? 0 ⎢ 1 1 1 ? ⎥ ⎣ ? 1 1 1 ⎦ (1958) 0 ? 1 1 Initializing unknowns to 0, by alternating projection we find the constrained matrix closest to the positive semidefinite cone, ⎡ ⎤ 1 1 0.5454 0 ⎢ 1 1 1 0.5454 ⎥ ⎣ 0.5454 1 1 1 ⎦ (1959) 0 0.5454 1 1 and we find the positive semidefinite matrix closest to the affine subset A (1949): ⎡ ⎤ 1.0521 0.9409 0.5454 0.0292 ⎢ 0.9409 1.0980 0.9451 0.5454 ⎥ ⎣ 0.5454 0.9451 1.0980 0.9409 ⎦ (1960) 0.0292 0.5454 0.9409 1.0521 These matrices (1959) and (1960) attain the Euclidean distance dist(A , S+) n . Convergence is illustrated in Figure 147.
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- Page 721 and 722: Bibliography [1] Suliman Al-Homidan
- Page 723 and 724: BIBLIOGRAPHY 723 [24] Alexander I.
- Page 725 and 726: BIBLIOGRAPHY 725 [52] Stephen Boyd,
- Page 727 and 728: BIBLIOGRAPHY 727 [78] Frank Critchl
- Page 729 and 730: BIBLIOGRAPHY 729 [105] Richard L. D
- Page 731 and 732: BIBLIOGRAPHY 731 [132] Michel X. Go
- Page 733 and 734: BIBLIOGRAPHY 733 [162] T. Herrmann,
- Page 735 and 736: BIBLIOGRAPHY 735 [191] Mark Kahrs a
- Page 737 and 738: BIBLIOGRAPHY 737 [220] K. V. Mardia
- Page 739 and 740: BIBLIOGRAPHY 739 [250] Pythagoras P
- Page 741 and 742: BIBLIOGRAPHY 741 [277] Anthony Man-
- Page 743 and 744: BIBLIOGRAPHY 743 [306] Michael W. T
- Page 745 and 746: [333] Margaret H. Wright. The inter
- Page 747 and 748: Index 0-norm, 203, 261, 294, 296, 2
E.10. ALTERNATING PROJECTION 699<br />
10 0<br />
dist(P 2 X i , S n +)<br />
10 −1<br />
0 2 4 6 8 10 12 14 16 18<br />
i<br />
Figure 147: Distance (confer (1953)) between PSD cone and iterate (1956) in<br />
affine subset A (1949) for Laurent’s completion problem; initially, decreasing<br />
geometrically.<br />
Laurent gives a problem for which no positive semidefinite completion<br />
exists: [203]<br />
⎡ ⎤<br />
1 1 ? 0<br />
⎢ 1 1 1 ?<br />
⎥<br />
⎣ ? 1 1 1 ⎦ (1958)<br />
0 ? 1 1<br />
Initializing unknowns to 0, by alternating projection we find the constrained<br />
matrix closest to the positive semidefinite cone,<br />
⎡<br />
⎤<br />
1 1 0.5454 0<br />
⎢ 1 1 1 0.5454<br />
⎥<br />
⎣ 0.5454 1 1 1 ⎦ (1959)<br />
0 0.5454 1 1<br />
and we find the positive semidefinite matrix closest to the affine subset A<br />
(1949): ⎡<br />
⎤<br />
1.0521 0.9409 0.5454 0.0292<br />
⎢ 0.9409 1.0980 0.9451 0.5454<br />
⎥<br />
⎣ 0.5454 0.9451 1.0980 0.9409 ⎦ (1960)<br />
0.0292 0.5454 0.9409 1.0521<br />
These matrices (1959) and (1960) attain the Euclidean distance dist(A , S+) n .<br />
Convergence is illustrated in Figure 147.
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