Raise the LSQR penalty cap to 1e6

[timholy]

On sparse nonsymmetric lognormal matrices cover_transversal was 1.5 to 29 times slower than cover_min, the ratio growing with size. Its multiplier iteration contracted by less than a factor of two per update once the penalty reached the LSQR cap of 1e5, stalled with a KKT residual of 1e-6 to 1e-4, and left the active-set finish tens to hundreds of steps.

The cause is the shape of the active set. At the minimizer the active constraints of the transversal-tight problem form one large tree: on a 10000x10000 matrix with six entries per row, 2792 rows with diameter 163 and Laplacian Fiedler value 2.6e-5, against trees of at most 41 nodes and Fiedler value 1.3e-2 for cover_min on the same matrix. The method of multipliers contracts at a rate near 1/(1 + κ s), with s bounded by that eigenvalue, so κ = 1e5 barely contracts while 1e6 gains a factor of two to four per update. Neither problem is degenerate: both have a clean split between active and slack constraints and one positive multiplier per active constraint.

With the cap at 1e6, cover_transversal certifies with 0 to 2 finish steps instead of 10 to 119 on four such matrices (three at n = 10000, one at n = 31623), in equal or shorter time; at n = 31623 the time drops from 14.2 s to 8.8 s. cover_min is unchanged within three solves. At 1e7 the iteration levels off near 1e-7, as does the exact dense path, so 1e6 is the usable ceiling in Float64.

The drain test moves to a banded matrix whose slack multipliers still need a penalty above the new cap. The cross-check of the symmetric finish against the asymmetric solve loosens to 1e-9: at the higher cap the asymmetric multiplier iteration can meet its own KKT tolerance and skip the finish, which leaves its scaled entries about 1e-10 from the certified minimizer.

Assisted-by: Claude Fable 5.1 noreply@anthropic.com

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 98.79%. Comparing base (67fd0a8) to head (34fcc13).

Additional details and impacted files

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  Coverage   98.79%   98.79%           
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  Files          23       23           
  Lines        6286     6286           
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  Hits         6210     6210           
  Misses         76       76           

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