KKT systems

AbstractKKTSystem{T, VT<:AbstractVector{T}, MT<:AbstractMatrix{T}, QN<:AbstractHessian{T}}

Abstract type for KKT system.

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

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
• $X = diag(x)$
• $S = diag(s)$
• $V = diag(ν)$
• $W = diag(w)$

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

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
• $Σₓ = X⁻¹ V$
• $Σₛ = S⁻¹ W$

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

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
• $Σₓ = X⁻¹ V$
• $Σₛ = S⁻¹ W$

create_kkt_system(
    ::Type{KKT},
    cb::AbstractCallback,
    ind_cons::NamedTuple,
    linear_solver::Type{LinSol};
    opt_linear_solver=default_options(linear_solver),
    hessian_approximation=ExactHessian,
) where {KKT<:AbstractKKTSystem, LinSol<:AbstractLinearSolver}

Instantiate a new KKT system with type KKT, associated to the the nonlinear program encoded inside the callback cb. The NamedTuple ind_cons stores the indexes of all the variables and constraints in the callback cb. In addition, the user should pass the linear solver linear_solver that will be used to solve the KKT system after it has been assembled.

Number of primal variables (including slacks) associated to the KKT system.

get_kkt(kkt::AbstractKKTSystem)::AbstractMatrix

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

Get Jacobian matrix

Get Hessian matrix

initialize!(kkt::AbstractKKTSystem)

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

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.

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).

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

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

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.

Nonzero in Jacobian

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

Implement the AbstractReducedKKTSystem in sparse COO format.

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

Implement the AbstractUnreducedKKTSystem in sparse COO format.

SparseCondensedKKTSystem{T, VT, MT, QN} <: AbstractCondensedKKTSystem{T, VT, MT, QN}

Implement the AbstractCondensedKKTSystem in sparse COO format.

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

Implement AbstractReducedKKTSystem with dense matrices.

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

DenseCondensedKKTSystem{T, VT, MT, QN} <: AbstractCondensedKKTSystem{T, VT, MT, QN}

Implement AbstractCondensedKKTSystem with dense matrices.

Requires a dense linear solver to factorize the associated KKT system (otherwise an error is returned).

AbstractKKTVector{T, VT}

Supertype for KKT's right-hand-side vectors $(x, s, y, z, ν, w)$.

number_primal(X::AbstractKKTVector)

Get total number of primal values $(x, s)$ in KKT vector X.

number_dual(X::AbstractKKTVector)

Get total number of dual values $(y, z)$ in KKT vector X.

full(X::AbstractKKTVector)

Return the all the values stored inside the KKT vector X.

primal(X::AbstractKKTVector)

Return the primal values $(x, s)$ stored in the KKT vector X.

dual(X::AbstractKKTVector)

Return the dual values $(y, z)$ stored in the KKT vector X.

primal_dual(X::AbstractKKTVector)

Return both the primal and the dual values $(x, s, y, z)$ stored in the KKT vector X.

dual_lb(X::AbstractKKTVector)

Return the dual values $ν$ associated to the lower-bound stored in the KKT vector X.

dual_ub(X::AbstractKKTVector)

Return the dual values $w$ associated to the upper-bound stored in the KKT vector X.

UnreducedKKTVector{T, VT<:AbstractVector{T}} <: AbstractKKTVector{T, VT}

Full KKT vector $(x, s, y, z, ν, w)$, associated to a AbstractUnreducedKKTSystem.