Add set for low-rank constrained SDP

Author: blegat

docs/src/background/duality.md

@@ -113,7 +113,9 @@ and similarly, the dual is:
 
 The scalar product is different from the canonical one for the sets
 [`PositiveSemidefiniteConeTriangle`](@ref), [`LogDetConeTriangle`](@ref),
-[`RootDetConeTriangle`](@ref).
+[`RootDetConeTriangle`](@ref),
+[`FrobeniusProductPostiviveSemidefiniteConeTriangle`](@ref) and
+[`LinearMatrixInequalityConeTriangle`](@ref).
 
 If the set ``C_i`` of the section [Duality](@ref) is one of these three cones,
 then the rows of the matrix ``A_i`` corresponding to off-diagonal entries are

docs/src/manual/standard_form.md

@@ -98,7 +98,8 @@ The matrix-valued set types implemented in MathOptInterface.jl are:
 | [`NormSpectralCone(r, c)`](@ref MathOptInterface.NormSpectralCone)    | ``\{ (t, X) \in \mathbb{R}^{1 + r \times c} : t \ge \sigma_1(X), X \mbox{ is a } r\times c\mbox{ matrix} \}``
 | [`NormNuclearCone(r, c)`](@ref MathOptInterface.NormNuclearCone)      | ``\{ (t, X) \in \mathbb{R}^{1 + r \times c} : t \ge \sum_i \sigma_i(X), X \mbox{ is a } r\times c\mbox{ matrix} \}`` |
 | [`HermitianPositiveSemidefiniteConeTriangle(d)`](@ref MathOptInterface.HermitianPositiveSemidefiniteConeTriangle) | The cone of Hermitian positive semidefinite matrices, with
-`side_dimension` rows and columns. |
+| [`FrobeniusProductPostiviveSemidefiniteConeTriangle(d, A)`](@ref MathOptInterface.FrobeniusProductPostiviveSemidefiniteConeTriangle) | The cone of positive semidefinite matrices, with `side_dimension` rows and columns and their Frobenius inner product with the matrices in `A`. |
+| [`LinearMatrixInequalityConeTriangle(d, A)`](@ref MathOptInterface.LinearMatrixInequalityConeTriangle) | The cone of vector `y` and symmetric `C`, with `side_dimension` rows and columns such that ``\sum_i y_i A_i + C`` is positive semidefinite. |
 
 Some of these cones can take two forms: `XXXConeTriangle` and `XXXConeSquare`.
 

docs/src/reference/standard_form.md

@@ -152,4 +152,6 @@ LogDetConeTriangle
 LogDetConeSquare
 RootDetConeTriangle
 RootDetConeSquare
+FrobeniusProductPostiviveSemidefiniteConeTriangle
+LinearMatrixInequalityConeTriangle
 ```

src/Utilities/model.jl

@@ -795,6 +795,8 @@ const LessThanIndicatorZero{T} =
         MOI.ScaledPositiveSemidefiniteConeTriangle,
         MOI.RootDetConeTriangle,
         MOI.RootDetConeSquare,
+        MOI.FrobeniusProductPostiviveSemidefiniteConeTriangle,
+        MOI.LinearMatrixInequalityConeTriangle,
         MOI.LogDetConeTriangle,
         MOI.LogDetConeSquare,
         MOI.AllDifferent,

src/Utilities/results.jl

@@ -575,6 +575,19 @@ function set_dot(
     return x[1] * y[1] + x[2] * y[2] + triangle_dot(x, y, set.side_dimension, 2)
 end
 
+function set_dot(
+    x::AbstractVector,
+    y::AbstractVector,
+    set::Union{
+        MOI.FrobeniusProductPostiviveSemidefiniteConeTriangle,
+        MOI.LinearMatrixInequalityConeTriangle,
+    },
+)
+    m = length(set.matrices)
+    return LinearAlgebra.dot(view(x, 1:m), view(y, 1:m)) +
+           triangle_dot(x, y, set.side_dimension, m)
+end
+
 """
     dot_coefficients(a::AbstractVector, set::AbstractVectorSet)
 
@@ -633,3 +646,15 @@ function dot_coefficients(a::AbstractVector, set::MOI.LogDetConeTriangle)
     triangle_coefficients!(b, set.side_dimension, 2)
     return b
 end
+
+function set_dot(
+    a::AbstractVector,
+    set::Union{
+        MOI.FrobeniusProductPostiviveSemidefiniteConeTriangle,
+        MOI.LinearMatrixInequalityConeTriangle,
+    },
+)
+    b = copy(a)
+    triangle_coefficients!(b, set.side_dimension, length(set.matrices))
+    return b
+end

src/sets.jl

@@ -1735,6 +1735,61 @@ function triangular_form(set::RootDetConeSquare)
     return RootDetConeTriangle(set.side_dimension)
 end
 
+"""
+    FrobeniusProductPostiviveSemidefiniteConeTriangle{M}(side_dimension::Int, matrices::Vector{M})
+
+Given `m` symmetric matrices `A_1`, ..., `A_m` given in `matrices`, the frobenius inner
+product of positive semidefinite matrices is the convex cone:
+``\\{ ((\\langle A_1, X \\rangle, ..., \\langle A_m, X \\rangle, X) \\in \\mathbb{R}^{m + d(d+1)/2} : X \\text{ postive semidefinite} \\}``,
+where the matrix `X` is represented in the same symmetric packed format as in
+the [`PositiveSemidefiniteConeTriangle`](@ref).
+"""
+struct FrobeniusProductPostiviveSemidefiniteConeTriangle{M} <: AbstractVectorSet
+    side_dimension::Int
+    matrices::Vector{M}
+end
+
+function dimension(s::FrobeniusProductPostiviveSemidefiniteConeTriangle)
+    return length(s.matrices) + s.side_dimension^2
+end
+
+function dual_set(s::FrobeniusProductPostiviveSemidefiniteConeTriangle)
+    return LinearMatrixInequalityConeTriangle(s.side_dimension, s.matrices)
+end
+
+function dual_set_type(
+    ::Type{FrobeniusProductPostiviveSemidefiniteConeTriangle{M}},
+) where {M}
+    return LinearMatrixInequalityConeTriangle
+end
+
+"""
+    LinearMatrixInequalityConeTriangle{M}(side_dimension::Int, matrices::Vector{M})
+
+Given `m` symmetric matrices `A_1`, ..., `A_m` given in `matrices`, the linear
+matrix inequality cone is the convex cone:
+``\\{ ((y, C) \\in \\mathbb{R}^{m + d(d+1)/2} : \\sum_{i=1}^m y_i A_i + C \\text{ postive semidefinite} \\}``,
+where the matrix `C` is represented in the same symmetric packed format as in
+the [`PositiveSemidefiniteConeTriangle`](@ref).
+"""
+struct LinearMatrixInequalityConeTriangle{M} <: AbstractVectorSet
+    side_dimension::Int
+    matrices::Vector{M}
+end
+
+dimension(s::LinearMatrixInequalityConeTriangle) = length(s.matrices) + s.side_dimension^2
+
+function dual_set(s::LinearMatrixInequalityConeTriangle)
+    return FrobeniusProductPostiviveSemidefiniteConeTriangle(
+        s.side_dimension,
+        s.matrices,
+    )
+end
+
+function dual_set_type(::Type{LinearMatrixInequalityConeTriangle{M}}) where {M}
+    return FrobeniusProductPostiviveSemidefiniteConeTriangle
+end
+
 """
     SOS1{T<:Real}(weights::Vector{T})