Linear algebra re-write #34
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* Remove commented code in HSD algorithm and related files * Remove obsolete functions in LinearAlgebra module * Update tests to reflect new conventions * De-activate multiple-precision tests (temporary) * De-activate UnitBlockAngular tests (temporary)
* Float32 is hardly used anyway * Rational breaks type stability, since `sqrt(::Rational)` returns a `Float64`
* Computation of residuals * Stopping criterion * Computation of h0 in the Newton system solve
Codecov Report
@@ Coverage Diff @@
## master #34 +/- ##
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- Coverage 84.91% 77.16% -7.76%
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Files 20 22 +2
Lines 1472 1673 +201
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+ Hits 1250 1291 +41
- Misses 222 382 +160
Continue to review full report at Codecov.
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* Solve proximal problem obtained from the homogeneous self-dual form * Remove unused proximal points * Update code for Newton system * Remove looser criteria for infeasibility detection * Remove log of tau and kappa
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The goal of this PR is to make the linear algebra layer considerably more flexible.
Linear algebra layers
AbstractLinearSolverPrior to this PR, the Newton system is systematically reduced to the normal equations, which are solved using a Cholesky factorization of the PSD matrix
A*D*A'.In this PR, we introduce the
AbstractLinearSolvertype, which --among others-- makes it easier to choose between:Only the resolution of the augmented system is exposed to the algorithm. The reduction to the normal equations, if applicable, is done within the linear solver object itself. See the docs for more details.
For sparse constraint matrix (and
Float32/Float64precision), the default setting is to use an LDL factorization of the (quasi-definite) augmented system. This approach better handles dense columns in the constraint matrix, and tends to be more numerically stable.For dense matrices, the reduction to the normal equations is automatic, and BLAS/LAPACK are used when applicable.
These settings are currently not exposed, i.e., one must modify the source code manually.
Regularizations
The augmented system
is symmetric indefinite, thus its LDL factorization may not exist.
Therefore, we introduce primal and dual regularizations
RpandRd, which make the system quasi-definite:Here,
RpandRdare diagonal matrices with positive diagonals, and any implementation ofAbstractLinearSolveris expected to be able to handle them explicitly.Future changes
The current interface for
AbstractLinearSolveris expected to change in the near future, with the introduction of traits and extra parameters to make it more user-friendly and (even more) flexible.Regularized algorithm
Given that we use black-box linear solvers, which does not allow explicit control over the factorization process (when a direct method is employed), the regularizations are handled in the IPM algorithm directly.
The homogeneous self-dual algorithm was modified accordingly, including updated stopping criteria. See paper for details.