Linear solvers

We suppose that the KKT system has been assembled previously into a given AbstractKKTSystem. Then, it remains to compute the Newton step by solving the KKT system for a given right-hand-side (given as a AbstractKKTVector). That's exactly the role of the linear solver.

If we do not assume any structure, the KKT system writes in generic form

\[K x = b\]

with $K$ the KKT matrix and $b$ the current right-hand-side. MadNLP provides a suite of specialized linear solvers to solve the linear system.

Inertia detection

If the matrix $K$ has negative eigenvalues, we have no guarantee that the solution of the KKT system is a descent direction with regards to the original nonlinear problem. That's the reason why most of the linear solvers compute the inertia of the linear system when factorizing the matrix $K$. The inertia counts the number of positive, negative and zero eigenvalues in the matrix. If the inertia does not meet a given criteria, then the matrix $K$ is regularized by adding a multiple of the identity to it: $K_r = K + \alpha I$.

Note

We recall that the inertia of a matrix $K$ is given as a triplet $(n,m,p)$, with $n$ the number of positive eigenvalues, $m$ the number of negative eigenvalues and $p$ the number of zero eigenvalues.

Factorization algorithm

In nonlinear programming, it is common to employ a LBL factorization to decompose the symmetric indefinite matrix $K$, as this algorithm returns the inertia of the matrix directly as a result of the factorization.

Note

When MadNLP runs in inertia-free mode, the algorithm does not require to compute the inertia when factorizing the matrix $K$. In that case, MadNLP can use a classical LU or QR factorization to solve the linear system $Kx = b$.

Solving a KKT system with MadNLP

We suppose available a AbstractKKTSystem kkt, properly assembled following the procedure presented previously. We can query the assembled matrix $K$ as

K = MadNLP.get_kkt(kkt)

6×6 SparseArrays.SparseMatrixCSC{Float64, Int32} with 13 stored entries:
 2.0     ⋅     ⋅     ⋅    ⋅    ⋅ 
 0.0  200.0    ⋅     ⋅    ⋅    ⋅ 
  ⋅      ⋅    0.0    ⋅    ⋅    ⋅ 
  ⋅      ⋅     ⋅    0.0   ⋅    ⋅ 
 0.0    0.0  -1.0    ⋅   0.0   ⋅ 
 1.0    0.0    ⋅   -1.0   ⋅   0.0

Then, if we want to pass the KKT matrix K to Lapack, this translates to

linear_solver = LapackCPUSolver(K)

LapackCPUSolver{Float64, SparseArrays.SparseMatrixCSC{Float64, Int32}}(sparse(Int32[1, 2, 5, 6, 2, 5, 6, 3, 5, 4, 6, 5, 6], Int32[1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6], [2.0, 0.0, 0.0, 1.0, 200.0, 0.0, 0.0, 0.0, -1.0, 0.0, -1.0, 0.0, 0.0], 6, 6), [1.9e-322 1.6e-322 … 6.4e-323 3.5e-323; 1.7e-322 1.5e-322 … 7.0e-323 4.0e-323; … ; 1.8e-322 1.4e-322 … 5.0e-323 2.5e-323; 1.63e-322 1.33e-322 … 4.4e-323 2.0e-323], 6, Float64[], Float64[], Float64[], [384.0, 5.20927505e-315, 0.0, 0.0, 2.1219958033e-314, 5.22339169e-315, 0.0, 6.34437920388833e-310, 0.0, 0.0  …  5.139594006e-315, 5.179930474e-315, 3.195172906639809e160, 7.873093353750373e160, 1.659495000339272e-76, 0.0, 0.0, 6.37e-322, 5.194387863e-315, 6.34438301122855e-310], 384, [0], -1, Base.RefValue{Int64}(0), [1, 128410107982032, 0, 0, 0, 0], MadNLP.LapackOptions(MadNLP.BUNCHKAUFMAN), MadNLP.MadNLPLogger(MadNLP.INFO, MadNLP.INFO, nothing))

The instance linear_solver does not copy the matrix $K$ and instead keep a reference to it.

linear_solver.A === K

true

That way every time we re-assemble the matrix $K$ in kkt, the values are directly updated inside linear_solver.

To compute the factorization inside linear_solver, one simply as to call:

MadNLP.factorize!(linear_solver)

LapackCPUSolver{Float64, SparseArrays.SparseMatrixCSC{Float64, Int32}}(sparse(Int32[1, 2, 5, 6, 2, 5, 6, 3, 5, 4, 6, 5, 6], Int32[1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6], [2.0, 0.0, 0.0, 1.0, 200.0, 0.0, 0.0, 0.0, -1.0, 0.0, -1.0, 0.0, 0.0], 6, 6), [2.0 0.0 … 0.0 0.0; 0.0 200.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 0.5 0.0 … -1.0 -0.5], 6, Float64[], Float64[], Float64[], [384.0, 5.20927505e-315, 0.0, 0.0, 2.1219958033e-314, 5.22339169e-315, 0.0, 6.34437920388833e-310, 0.0, 0.0  …  5.139594006e-315, 5.179930474e-315, 3.195172906639809e160, 7.873093353750373e160, 1.659495000339272e-76, 0.0, 0.0, 6.37e-322, 5.194387863e-315, 6.34438301122855e-310], 384, [0], -1, Base.RefValue{Int64}(0), [1, 2, -5, -5, -6, -6], MadNLP.LapackOptions(MadNLP.BUNCHKAUFMAN), MadNLP.MadNLPLogger(MadNLP.INFO, MadNLP.INFO, nothing))

Once the factorization computed, computing the backsolve for a right-hand-side b amounts to

nk = size(kkt, 1)
b = rand(nk)
MadNLP.solve_linear_system!(linear_solver, b)

6-element Vector{Float64}:
  0.7639387793724377
  0.0007307673763606448
 -0.5242668621037749
  0.44766912750494253
 -0.0023472310470741276
 -0.6729096880541992

The values of b being modified inplace to store the solution $x$ of the linear system $Kx =b$.