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: $K_r = K + \alpha I$.
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 Bunch-Kaufman factorization (or LDL factorization) to factorize the matrix $K$, as this algorithm returns the inertia of the matrix directly as a result of the factorization.
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 Matrix{Float64}:
2.0 0.0 0.0 0.0 0.0 0.0
0.0 200.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 -1.0 0.0 0.0 0.0
1.0 0.0 0.0 -1.0 0.0 0.0
Then, if we want to pass the KKT matrix K
to Lapack, this translates to
linear_solver = LapackCPUSolver(K)
LapackCPUSolver{Float64}([2.0 0.0 … 0.0 0.0; 0.0 200.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 1.0 0.0 … 0.0 0.0], [2.0 0.0 … 0.0 0.0; 0.0 200.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 1.0 0.0 … 0.0 0.0], [0.0], -1, Base.RefValue{Int64}(0), Dict{Symbol, Any}(), 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.dense === 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}([2.0 0.0 … 0.0 0.0; 0.0 200.0 … 0.0 0.0; … ; 0.0 0.0 … 0.0 0.0; 1.0 0.0 … 0.0 0.0], [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], [384.0, 0.0, 5.58679274e-316, 5.58662713e-316, 1.2875975e-316, 0.0, 6.365987414e-314, 0.0, 0.0, 0.0 … 2.4747693e-316, 0.0, 4.2439918645e-314, 0.0, 0.0, 0.0, 5.32189727e-316, 0.0, 1.2875833e-316, 1.2875833e-316], 384, Base.RefValue{Int64}(0), Dict{Symbol, Any}(:ipiv => [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_solver, b)
6-element Vector{Float64}:
0.631358073804925
0.0040793963052856005
-0.2796567800246169
-0.18552533773316882
-0.0331186942806706
-0.6074504205132841
The values of b
being modified inplace to store the solution $x$ of the linear system $Kx =b$.