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Improve the linear solver CholmodSQD #92

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Mar 24, 2021
Merged

Improve the linear solver CholmodSQD #92

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35 changes: 24 additions & 11 deletions src/KKT/cholmod.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -56,9 +56,10 @@ mutable struct CholmodSQD <: CholmodSolver

# Problem data
A::SparseMatrixCSC{Float64}
θ::Vector{Float64}
θ::Vector{Float64} # diagonal scaling
regP::Vector{Float64} # primal-dual regularization
regD::Vector{Float64} # dual regularization
ξ::Vector{Float64} # right-hand side of the augmented system

# Left-hand side matrix
S::SparseMatrixCSC{Float64, Int} # TODO: use `CHOLMOD.Sparse` instead
Expand All @@ -70,6 +71,9 @@ mutable struct CholmodSQD <: CholmodSolver
function CholmodSQD(A::AbstractMatrix{Float64})
m, n = size(A)
θ = ones(Float64, n)
regP = ones(Float64, n)
regD = ones(Float64, m)
ξ = zeros(Float64, n+m)

S = [
spdiagm(0 => -θ) A';
Expand All @@ -79,7 +83,7 @@ mutable struct CholmodSQD <: CholmodSolver
# TODO: Symbolic factorization only
F = ldlt(Symmetric(S))

return new(m, n, A, θ, ones(Float64, n), ones(Float64, m), S, F)
return new(m, n, A, θ, regP, regD, ξ, S, F)
end

end
Expand Down Expand Up @@ -148,15 +152,24 @@ function solve!(
)
m, n = kkt.m, kkt.n

# Set-up right-hand side
ξ = [ξd; ξp]
# Set-up the right-hand side
@inbounds for i in 1:n
kkt.ξ[i] = ξd[i]
end
@inbounds for i in 1:m
kkt.ξ[i+n] = ξp[i]
end

# Solve augmented system
d = kkt.F \ ξ
d = kkt.F \ kkt.ξ

# Recover dx, dy
@views dx .= d[1:n]
@views dy .= d[(n+1):(m+n)]
@inbounds for i in 1:n
dx[i] = d[i]
end
@inbounds for i in 1:m
dy[i] = d[i+n]
end

# Iterative refinement
# TODO:
Expand Down Expand Up @@ -186,15 +199,15 @@ end

Linear solver for the 2x2 augmented system
```
[-(Θ⁻¹ + Rp) Aᵀ] [dx] = [xi_d]
[ A Rd] [dy] [xi_p]
[-(Θ⁻¹ + Rp) Aᵀ] [dx] = [ξd]
[ A Rd] [dy] [ξp]
```
with ``A`` sparse.

Uses a Cholesky factorization of the positive definite normal equations system
```
(A * ((Θ⁻¹ + Rp)⁻¹ * Aᵀ + Rd) dy = xi_p + A * (θ⁻¹ + Rp)⁻¹ * xi_d
dx = (Θ⁻¹ + Rp)⁻¹ * (Aᵀ * dy - xi_d)
(A * ((Θ⁻¹ + Rp)⁻¹ * Aᵀ + Rd) dy = ξp + A * (θ⁻¹ + Rp)⁻¹ * ξd
dx = (Θ⁻¹ + Rp)⁻¹ * (Aᵀ * dy - ξd)
```
"""
mutable struct CholmodSPD <: CholmodSolver
Expand Down