Change scaling in HiPO

[filikat]

• Currently, HiPO uses Curtis-Reid scaling to scale the problem before solving it.
  This PR changes the scaling to infinity-norm equilibration of the rows and columns of the matrix. The scaling procedure takes into account also the magnitude of the diagonal of matrix Q and the width of the bounds interval.
  Changing the scaling significantly reduces the number of iterations on some instances, including some of the PyPSA problems, and improves robustness in general.
• Some small pathological tests on infeasible problems occasionally fail with HiPO, due to numerical inaccuracies. These tests are not very useful, since they are not representative of real problems, and their failure is not symptomatic of an error in a commit, but rather of a random numerical perturbation that pushes the test to fail. They have been removed and substituted with tests on more realistic infeasible instances.
• Printing of the ranges of A, b, c and bounds has been slightly changed and made less verbose. Dev printing now shows the ranges of inf and one norms of rows and columns of A.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 81.22%. Comparing base (e3e7bf2) to head (544ac5e).
⚠️ Report is 168 commits behind head on latest.

Additional details and impacted files

@@            Coverage Diff             @@
##           latest    #2934      +/-   ##
==========================================
- Coverage   81.54%   81.22%   -0.33%     
==========================================
  Files         347      358      +11     
  Lines       86942    88576    +1634     
==========================================
+ Hits        70901    71943    +1042     
- Misses      16041    16633     +592     

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[jajhall]

So, you're doing ten rounds of scaling, and then moving the factors to the nearest power-of-2. Maybe they should be moved to the nearest power-of-2 after each scaling round, so that convergence is "over the allowable values"

Otherwise, this looks fine

[filikat]

I thought about this, but the results are actually a bit worse that way.
My intuition is that by pushing the values to the nearest power of 2 after each iteration, some scaling coefficients may be prevented from reaching a better value that would take many small steps.

[jajhall]

I thought about this, but the results are actually a bit worse that way.
My intuition is that by pushing the values to the nearest power of 2 after each iteration, some scaling coefficients may be prevented from reaching a better value that would take many small steps.

Fair point