IPM

AbstractCone

A convex cone.

CofreeCone <: AbstractCone

The cone of all n-dimensional Euclidean vectors.

ExponentialCone <: AbstractTDCone

The exponential cone, consisting of all triples (x, y, z) such that x > 0, y > 0, and y log(x/y) ≥ z.

IPMProblem(Q, B, c, g, K)

Construct an IPMProblem.

PositiveCone <: AbstractCone

An n-dimensional positive orthant.

PowerCone{T} <: AbstractTDCone

The three-dimensional power cone with parameter α ∈ (0, 1), consisting of all triples (x₁, x₂, x₃) such that x₁ ≥ 0, x₂ ≥ 0, and x₁^α x₂^(1-α) ≥ |x₃|.

Scaling{T}(n, m)

Construct an identity (trivial) scaling with all entries equal to 1.

SecondOrderCone <: AbstractCone

The n + 1-dimensional second-order cone, consisting of all pairs (x, y) such that y ≥ ‖x‖.

SemidefiniteCone <: AbstractCone

A cone of n × n positive-semidefinite matrices.

cache(caches, i, cone)

Get the ith cache.

cachesize(cone, n)

Return the number of cache slots needed for a cone with embedding dimension n.

corr!(r, p, d, Δp, Δd, σμ, cache)

Set r to the Mehrotra corrector term r = -d - σμ f'(p) - η, where η is the third-order correction η = -½ f'''(p)[Δp, f''(p)⁻¹ Δd]. If p and d are elements of a symmetric cone, this formula simplifies to r = -d + (σμ e - Δp ∘ Δd) / p, where Δp ∘ Δd is the Jordan product of Δp and Δd.

degree(cone::AbstractCone, n::Integer)

Get the rank of a cone with embedding dimension n.

equilibrate!(scaling, B, Q, c, g; itmax=10, tol=1e-3)

Run block-aware Ruiz equilibration on the conic-QP data in place. Mutates scaling, B, Q, c, and g. Returns scaling.

Keyword arguments:

• itmax : maximum Ruiz sweeps.
• tol : stop when every row/block ∞-norm is within tol of 1.

identity!(x::AbstractVector, cone::AbstractCone)

Set x to the fixed point -f'(e) = e of the barrier.

initcache!(cache)

Initialise a cache.

maxsteps(p, Δp, d, Δd, cache)

Compute the largest numbers 0 < τp, τd ≤ 1 such that p + τp Δp and d + τd Δd lie in the interior of their respective cones

scale!(H, p, d, cache)

Set H to the Tuncel scaling matrix. If p and d are elements of a symmetric cone, this is the Hessian f''(w) of the barrier at the Nesterov-Todd scaling point w.

scale!(p, d, y, scaling)

Apply forward scaling to vectors in place:

p ← D⁻¹ p,   d ← D d,   y ← E⁻¹ y

unscale!(p, d, y, scaling)

Apply inverse scaling to vectors in place:

p ← D p,   d ← D⁻¹ d,   y ← E y

workspacesize(cone, n)

Return the number of workspace floats needed for a cone with embedding dimension n.

solve(problem::IPMProblem;
    verbose=true,
    step_frac=0.99,
    feas_tol=1e-8,
    gap_tol=1e-8,
    itmax=100,
    near_factor=1000.0,
    stall_tol=1e-6,
    forcing_ceil=0.3,
    forcing_frac=1.0,
    refine_itmax=10,
    refine_stall=0.5,
    scale_itmax=10,
    kkt=UzawaSettings(),
)

Solve an IPMProblem.