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Table 5 Residuals for the solutions returned by CDCS with \(\epsilon _{\text {tol}} = 10^{-3}\) and the maximum number of iterations fixed to 2000

From: Chordal decomposition in operator-splitting methods for sparse semidefinite programs

 

CDCS-primal

CDCS-dual

CDCS-hsde

\(\epsilon _{\mathrm {c}}\)

\(\epsilon _\alpha \)

\(\epsilon _{\mathrm {c}}\)

\(\epsilon _{\mathrm {d}}\)

\(\epsilon _\alpha \)

\(\epsilon _{\mathrm {p}}\)

\(\epsilon _{\mathrm {d}}\)

\(\epsilon _{\mathrm {c}}\)

maxG11

9.97e−4

2.72e−4

1.34e−4

2.48e−4

2.84e−4

5.52e−4

3.46e−4

9.98e−4

maxG32

9.96e−4

2.22e−4

4.75e−4

11.1e−4

1.78e−4

2.74e−4

2.68e−4

9.97e−4

qpG11

3.70e−4

3.62e−4

9.94e−4

20.0e−4

2.76e−4

9.66e−4

3.60e−4

8.40e−4

qpG51

9.34e−4

0.54e−4

9.91e−4

4.02e−4

0.15e−4

13.3e−4

31.3e−4

2.10e−4

rs35

9.99e−4

7.05e−4

6.41e−4

9.36e−4

0.01e−4

0.31e−4

13.6e−4

9.99e−4

rs200

9.92e−4

6.73e−4

1.48e−4

2.47e−4

2.74e−4

9.37e−4

9.95e−4

9.71e−4

rs228

9.97e−4

5.92e−4

2.14e−4

2.71e−4

1.75e−4

8.82e−4

9.83e−4

9.86e−4

rs365

9.98e−4

2.98e−4

4.61e−4

9.99e−4

0.36e−4

2.50e−4

9.99e−4

9.97e−4

rs1555

2.21e−4

1.55e−4

9.97e−4

19.6e−4

0.98e−4

11.0e−4

1.89e−4

9.99e−4

rs1907

9.98e−4

3.27e−4

6.24e−4

12.0e−4

0.23e−4

3.78e−4

9.96e−4

9.90e−4

  1. The residual \(\epsilon _{\mathrm {c}}\), defined in (23a), (31a), and (50) for CDCS-primal, CDCS-dual, and CDCS-hsde respectively, measures the error in the consensus constraints of the decomposed SDPs. The quantities \(\epsilon _{\mathrm {p}}\) and \(\epsilon _{\mathrm {d}}\) are defined in (49a) and (49b), respectively, and measure the primal-dual residuals of the equality constraints for the original SDPs (before decomposition). Finally, \(\epsilon _\alpha \) computed using (54) measures the violation of the semidefiniteness constraint in the primal SDP