AbstractCondensedKKTSystem{T, MT} <: AbstractKKTSystem{T, MT}

The condensed KKT system simplifies further the `AbstractReducedKKTSystem by removing the rows associated to the slack variables $s$ and the inequalities.

At the primal-dual iterate $(x, y)$, the matrix writes

[Wₓₓ + Σₓ + Aᵢ' Σₛ Aᵢ    Aₑ']  [Δx]
[         Aₑ              0 ]  [Δy]

with

• $Wₓₓ$: Hessian of the Lagrangian.
• $Aₑ$: Jacobian of the equality constraints
• $Aᵢ$: Jacobian of the inequality constraints
• $Σₓ = X⁻¹ V$
• $Σₛ = S⁻¹ W$

AbstractReducedKKTSystem{T, MT} <: AbstractKKTSystem{T, MT}

The reduced KKT system is a simplification of the original Augmented KKT system. Comparing to AbstractUnreducedKKTSystem), AbstractReducedKKTSystem removes the two last rows associated to the bounds' duals $(ν, w)$.

At a primal-dual iterate $(x, s, y, z)$, the matrix writes

[Wₓₓ + Σₓ   0    Aₑ'   Aᵢ']  [Δx]
[ 0         Σₛ    0    -I ]  [Δs]
[Aₑ         0     0     0 ]  [Δy]
[Aᵢ        -I     0     0 ]  [Δz]

with

• $Wₓₓ$: Hessian of the Lagrangian.
• $Aₑ$: Jacobian of the equality constraints
• $Aᵢ$: Jacobian of the inequality constraints
• $Σₓ = X⁻¹ V$
• $Σₛ = S⁻¹ W$

AbstractUnreducedKKTSystem{T, MT} <: AbstractKKTSystem{T, MT}

Augmented KKT system associated to the linearization of the KKT conditions at the current primal-dual iterate $(x, s, y, z, ν, w)$.

The associated matrix is

[Wₓₓ  0  Aₑ'  Aᵢ'  -I   0 ]  [Δx]
[ 0   0   0   -I    0  -I ]  [Δs]
[Aₑ   0   0    0    0   0 ]  [Δy]
[Aᵢ  -I   0    0    0   0 ]  [Δz]
[V    0   0    0    X   0 ]  [Δν]
[0    W   0    0    0   S ]  [Δw]

with

• $Wₓₓ$: Hessian of the Lagrangian.
• $Aₑ$: Jacobian of the equality constraints
• $Aᵢ$: Jacobian of the inequality constraints
• $X = diag(x)$
• $S = diag(s)$
• $V = diag(ν)$
• $W = diag(w)$

DenseKKTSystem{T, VT, MT} <: AbstractReducedKKTSystem{T, MT}

Implement AbstractReducedKKTSystem with dense matrices.

Requires a dense linear solver to be factorized (otherwise an error is returned).

SparseKKTSystem{T, MT} <: AbstractReducedKKTSystem{T, MT}

Implement the AbstractReducedKKTSystem in sparse COO format.

SparseUnreducedKKTSystem{T, MT} <: AbstractUnreducedKKTSystem{T, MT}

Implement the AbstractUnreducedKKTSystem in sparse COO format.

build_kkt!(kkt::AbstractKKTSystem)

Assemble the KKT matrix before calling the factorization routine.

compress_hessian!(kkt::AbstractKKTSystem)

Compress the Hessian inside kkt's internals. This function is called every time a new Hessian is evaluated.

Default implementation do nothing.

compress_jacobian!(kkt::AbstractKKTSystem)

Compress the Jacobian inside kkt's internals. This function is called every time a new Jacobian is evaluated.

By default, the function updates in the Jacobian the coefficients associated to the slack variables.

Get Hessian matrix

Get Jacobian matrix

get_kkt(kkt::AbstractKKTSystem)::AbstractMatrix

Return a pointer to the KKT matrix implemented in kkt. The pointer is passed afterward to a linear solver.

Dense Hessian callback

initialize!(kkt::AbstractKKTSystem)

Initialize KKT system with default values. Called when we initialize the InteriorPointSolver storing the current KKT system kkt.

is_inertia_correct(kkt::AbstractKKTSystem, n::Int, m::Int, p::Int)

Check if the inertia $(n, m, p)$ returned by the linear solver is adapted to the KKT system implemented in kkt.

Return true if KKT system is reduced.

Dense Jacobian callback

jtprod!(y::AbstractVector, kkt::AbstractKKTSystem, x::AbstractVector)

Multiply with transpose of Jacobian and store the result in y, such that $y = A' x$ (with $A$ current Jacobian).

Nonzero in Jacobian

Number of primal variables associated to the KKT system.

regularize_diagonal!(kkt::AbstractKKTSystem, primal_values::AbstractVector, dual_values::AbstractVector)

Regularize the values in the diagonal of the KKT system. Called internally inside the interior-point routine.

set_jacobian_scaling!(kkt::AbstractKKTSystem, scaling::AbstractVector)

Set the scaling of the Jacobian with the vector scaling storing the scaling for all the constraints in the problem.