Cones module

Proper cone definitions, oracles, and utilities.

abstract type Cone{T<:Real}

A proper cone.

vec_length(_, len)

The dimension of the real vectorization of a real or complex vector of length len::Int.

vec_copyto!(v1, v2)

Copy a vector in-place.

vec_copyto!(rvec, cvec)

Copy a complex vector to a real vector in-place.

vec_copyto!(cvec, rvec)

Copy a real vector to a complex vector in-place.

svec_length(side)

The dimension of the vectorized triangle of a real symmetric matrix with side dimension side::Int.

svec_length(_, side)

The dimension of the real vectorized triangle of a real symmetric or complex Hermitian matrix with side dimension side::Int.

svec_side(len)

The side dimension of a real symmetric matrix with vectorized triangle length len::Int.

svec_side(_, len)

The side dimension of a real symmetric or complex Hermitian matrix with real vectorized triangle length len::Int.

svec_idx(row, col)

The index in the vectorized triangle of a symmetric matrix for element (row::Int, col::Int).

block_idxs(incr, block)

The indices corresponding to block block::Int in a vector of blocks with equal length incr::Int.

scale_svec!(arr, scal)

Rescale the elements corresponding to off-diagonals in arr::AbstractVecOrMat, with scaling scal::Real, for real symmetric matrices.

scale_svec_incr!(arr, scal, incr)

Rescale the elements corresponding to off-diagonals in arr::AbstractVecOrMat, with scaling scal::Real and block increment incr::Int, for real symmetric matrices.

smat_to_svec!(vec, mat, rt2)

Copy a real symmetric matrix upper triangle to a svec-scaled vector in-place.

smat_to_svec!(vec, mat, rt2)

Copy a complex Hermitian matrix upper triangle to a svec-scaled real vector in-place.

svec_to_smat!(mat, vec, rt2)

Copy a svec-scaled vector to a real symmetric matrix upper triangle in-place.

svec_to_smat!(mat, vec, rt2)

Copy a svec-scaled real vector to a complex Hermitian matrix upper triangle in-place.

dimension(cone)

The real vector dimension of the cone.

get_nu(cone)

The barrier parameter $\nu$ of the cone.

set_initial_point!(arr, cone)

Set the array equal to the initial interior point for the cone.

is_feas(cone)

Returns true if and only if the currently-loaded primal point is strictly feasible for the cone.

is_dual_feas(cone)

Returns false only if the currently-loaded dual point is outside the interior of the cone's dual cone.

grad(cone)

The gradient of the cone's barrier function at the currently-loaded primal point.

hess(cone)

The Hessian (symmetric positive definite) of the cone's barrier function at the currently-loaded primal point.

inv_hess(cone)

The inverse Hessian (symmetric positive definite) of the cone's barrier function at the currently-loaded primal point.

hess_prod!(prod, arr, cone)

Compute the product of the Hessian of the cone's barrier function at the currently-loaded primal point with a vector or array, in-place.

inv_hess_prod!(prod, arr, cone)

Compute the product of the inverse Hessian of the cone's barrier function at the currently-loaded primal point with a vector or array, in-place.

use_dder3(_)

Returns true if and only if the oracle for the third-order directional derivative oracle dder3 can be computed.

dder3(cone, dir)

Compute the third-order directional derivative, in the direction dir, the cone's barrier function at the currently-loaded primal point.

mutable struct Nonnegative{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Nonnegative cone of dimension dim.

Nonnegative{T}(dim::Int)

mutable struct PosSemidefTri{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Real symmetric or complex Hermitian positive semidefinite cone of dimension dim in svec (scaled upper triangle) format.

PosSemidefTri{T, R}(dim::Int)

mutable struct DoublyNonnegativeTri{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Real symmetric doubly nonnegative cone (intersection of nonnegative and positive semidefinite cones) of dimension dim in svec format.

DoublyNonnegativeTri{T}(dim::Int, use_dual::Bool = false)

mutable struct PosSemidefTriSparse{I<:Hypatia.Cones.PSDSparseImpl, T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Real symmetric or complex Hermitian sparse positive semidefinite cone of side dimension side and sparse lower triangle row and column indices rows, cols in svec format. Note all diagonal elements must be present.

PosSemidefTriSparse{T, R}(side::Int, rows::Vector{Int}, cols::Vector{Int}, use_dual::Bool = false)

mutable struct LinMatrixIneq{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Linear matrix inequality cone parametrized by list of real symmetric or complex Hermitian matrices mats of equal dimension.

LinMatrixIneq{T}(mats::Vector, use_dual::Bool = false)

mutable struct EpiNormInf{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Epigraph of real or complex infinity norm cone of dimension dim.

EpiNormInf{T, R}(dim::Int, use_dual::Bool = false)

mutable struct EpiNormEucl{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Epigraph of Euclidean norm (AKA second-order) cone of dimension dim.

EpiNormEucl{T}(dim::Int)

mutable struct EpiPerSquare{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Epigraph of perspective function of halved squared Euclidean norm (AKA rotated second-order) cone of dimension dim.

EpiPerSquare{T}(dim::Int)

mutable struct EpiNormSpectralTri{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Epigraph of real symmetric or complex Hermitian matrix spectral norm (i.e. maximum absolute value of eigenvalues) cone of dimension dim in svec format.

EpiNormSpectralTri{T, R}(dim::Int, use_dual::Bool = false)

mutable struct EpiNormSpectral{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Epigraph of real or complex matrix spectral norm (i.e. maximum singular value) for a matrix (stacked column-wise) of nrows rows and ncols columns with nrows ≤ ncols.

EpiNormSpectral{T, R}(nrows::Int, ncols::Int, use_dual::Bool = false)

mutable struct MatrixEpiPerSquare{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Matrix epigraph of perspective function of real or complex matrix outer product for a matrix (stacked column-wise) of nrows rows and ncols columns with nrows ≤ ncols.

MatrixEpiPerSquare{T, R}(nrows::Int, ncols::Int, use_dual::Bool = false)

mutable struct GeneralizedPower{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Generalized power cone parametrized by powers α in the unit simplex and dimension d of the normed variables.

GeneralizedPower{T}(α::Vector{T}, d::Int, use_dual::Bool = false)

mutable struct HypoPowerMean{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Hypograph of weighted power mean cone parametrized by powers α in the unit simplex.

HypoPowerMean{T}(α::Vector{T}, use_dual::Bool = false)

mutable struct HypoGeoMean{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Hypograph of geometric mean cone of dimension dim.

HypoGeoMean{T}(dim::Int, use_dual::Bool = false)

mutable struct HypoRootdetTri{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Hypograph of real symmetric or complex Hermitian root-determinant cone of dimension dim in svec format.

HypoRootdetTri{T, R}(dim::Int, use_dual::Bool = false)

mutable struct HypoPerLog{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Hypograph of perspective function of sum-log cone of dimension dim.

HypoPerLog{T}(dim::Int, use_dual::Bool = false)

mutable struct HypoPerLogdetTri{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Hypograph of perspective function of real symmetric or complex Hermitian log-determinant cone of dimension dim in svec format.

HypoPerLogdetTri{T, R}(dim::Int, use_dual::Bool = false)

mutable struct EpiPerSepSpectral{Q<:Hypatia.Cones.ConeOfSquares, T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Epigraph of perspective function of a convex separable spectral function h over a cone of squares Q on a Jordan algebra with rank d.

EpiPerSepSpectral{Q, T}(h::Hypatia.Cones.SepSpectralFun, d::Int, use_dual::Bool = false)

mutable struct EpiRelEntropy{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Epigraph of vector relative entropy cone of dimension dim.

EpiRelEntropy{T}(dim::Int, use_dual::Bool = false)

mutable struct EpiTrRelEntropyTri{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Epigraph of matrix relative entropy cone of dimension dim in svec format.

EpiTrRelEntropyTri{T}(dim::Int, use_dual::Bool = false)

mutable struct WSOSInterpNonnegative{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.Cone{T<:Real}

Interpolant-basis weighted sum-of-squares polynomial cone of dimension U, for real or real-valued complex polynomials, parametrized by vector of matrices Ps derived from interpolant basis and polynomial domain constraints.

WSOSInterpNonnegative{T, R}(U::Int, Ps::Vector{Matrix{R}}, use_dual::Bool = false)

mutable struct WSOSInterpPosSemidefTri{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Interpolant-basis weighted sum-of-squares polynomial (of dimension U) positive semidefinite matrix (of side dimension R) cone, parametrized by vector of matrices Ps derived from interpolant basis and polynomial domain constraints.

WSOSInterpPosSemidefTri{T}(R::Int, U::Int, Ps::Vector{Matrix{T}}, use_dual::Bool = false)

mutable struct WSOSInterpEpiNormEucl{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Interpolant-basis weighted sum-of-squares polynomial (of dimension U) epigraph of ℓ₂ norm (of dimension R) cone, parametrized by vector of matrices Ps derived from interpolant basis and polynomial domain constraints.

WSOSInterpEpiNormEucl{T}(R::Int, U::Int, Ps::Vector{Matrix{T}}, use_dual::Bool = false)

mutable struct WSOSInterpEpiNormOne{T<:Real} <: Hypatia.Cones.Cone{T<:Real}

Interpolant-basis weighted sum-of-squares polynomial (of dimension U) epigraph of ℓ₁ norm (of dimension R) cone, parametrized by vector of matrices Ps derived from interpolant basis and polynomial domain constraints.

WSOSInterpEpiNormOne{T}(R::Int, U::Int, Ps::Vector{Matrix{T}}, use_dual::Bool = false)

abstract type PSDSparseImpl

An implementation type for the sparse positive semidefinite cone PosSemidefTriSparse.

struct PSDSparseDense <: Hypatia.Cones.PSDSparseImpl

Dense Cholesky-based implementation for the sparse positive semidefinite cone PosSemidefTriSparse.

struct PSDSparseCholmod <: Hypatia.Cones.PSDSparseImpl

CHOLMOD sparse Cholesky-based implementation for the sparse positive semidefinite cone PosSemidefTriSparse. Note only BLAS floating point types are supported.

abstract type ConeOfSquares{T<:Real}

A cone of squares on a Jordan algebra.

struct VectorCSqr{T<:Real} <: Hypatia.Cones.ConeOfSquares{T<:Real}

Real vector cone of squares.

struct MatrixCSqr{T<:Real, R<:Union{Complex{T<:Real}, T<:Real}} <: Hypatia.Cones.ConeOfSquares{T<:Real}

Real symmetric or complex Hermitian positive semidefinite cone of squares.

vector_dim(
    _::Type{<:Hypatia.Cones.VectorCSqr},
    d::Int64
) -> Int64

The rank of the vector cone of squares, equal to the vector length.

vector_dim(
    _::Type{<:Hypatia.Cones.MatrixCSqr{<:Real, R}},
    d::Int64
) -> Int64

The rank of the matrix cone of squares, equal to the side dimension of the matrix.

abstract type SepSpectralFun

A univariate convex function defined on positive reals.

struct NegLogSSF <: Hypatia.Cones.SepSpectralFun

The negative logarithm function $x \to - \log(x)$.

struct NegEntropySSF <: Hypatia.Cones.SepSpectralFun

The negative entropy function $x \to x \log(x)$.

struct NegSqrtSSF <: Hypatia.Cones.SepSpectralFun

The negative square root function $x \to -x^{1/2}$. Note this is a special case of the negative power: NegPower01SSF(0.5).

struct NegPower01SSF <: Hypatia.Cones.SepSpectralFun

The negative power function $x \to -x^p$ parametrized by $p \in (0, 1)$

struct Power12SSF <: Hypatia.Cones.SepSpectralFun

The power function $x \to x^p$ parametrized by $p \in (1, 2]$. Note for $p = 2$, it is more efficient to use EpiPerSquare.