Add set for low-rank constrained SDP

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),
+[`SetWithDotProducts`](@ref) and
+[`LinearCombinationInSet`](@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

@@ -82,6 +82,8 @@ The vector-valued set types implemented in MathOptInterface.jl are:
 | [`RelativeEntropyCone(d)`](@ref MathOptInterface.RelativeEntropyCone) | ``\{ (u, v, w) \in \mathbb{R}^{d} : u \ge \sum_i w_i \log (\frac{w_i}{v_i}), v_i \ge 0, w_i \ge 0 \}`` |
 | [`HyperRectangle(l, u)`](@ref MathOptInterface.HyperRectangle)        | ``\{x \in \bar{\mathbb{R}}^d: x_i \in [l_i, u_i] \forall i=1,\ldots,d\}`` |
 | [`NormCone(p, d)`](@ref MathOptInterface.NormCone)       | ``\{ (t,x) \in \mathbb{R}^{d} : t \ge \left(\sum\limits_i \lvert x_i \rvert^p\right)^{\frac{1}{p}} \}`` |
+| [`SetWithDotProcuts(s, v)`](@ref MathOptInterface.SetWithDotProducts) | The cone `s` with dot products with the fixed vectors `v`. |
+| [`LinearCombinationInSet(s, v)`](@ref MathOptInterface.LinearCombinationInSet) | The cone of vector `(y, x)` such that ``\sum_i y_i v_i + x`` belongs to `s`. |
 
 ## Matrix cones
 

docs/src/reference/standard_form.md

@@ -153,4 +153,6 @@ LogDetConeTriangle
 LogDetConeSquare
 RootDetConeTriangle
 RootDetConeSquare
+SetWithDotProducts,
+LinearCombinationInSet,
 ```

src/Utilities/set_dot.jl

@@ -86,6 +86,16 @@ 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.SetWithDotProducts,MOI.LinearCombinationInSet},
+)
+    m = length(set.matrices)
+    return LinearAlgebra.dot(view(x, 1:m), view(y, 1:m)) +
+           set_dot(view(x, (m+1):length(x)), view(y, (m+1):length(y)), set.set)
+end
+
 """
     dot_coefficients(a::AbstractVector, set::AbstractVectorSet)
 
@@ -145,6 +155,16 @@ function dot_coefficients(a::AbstractVector, set::MOI.LogDetConeTriangle)
     return b
 end
 
+function dot_coefficients(
+    a::AbstractVector,
+    set::Union{MOI.SetWithDotProducts,MOI.LinearCombinationInSet},
+)
+    b = copy(a)
+    m = length(set.vectors)
+    b[(m+1):end] = dot_coefficients(b[(m+1):end], set.set)
+    return b
+end
+
 # For `SetDotScalingVector`, we would like to compute the dot product
 # of canonical vectors in O(1) instead of O(n)
 # See https://github.com/jump-dev/Dualization.jl/pull/135

src/sets.jl

@@ -1791,6 +1791,50 @@ function Base.getproperty(
     end
 end
 
+"""
+    SetWithDotProducts(set, vectors::Vector{V})
+
+Given a set `set` of dimension `d` and `m` vectors `v_1`, ..., `v_m` given in `vectors`, this is the set:
+``\\{ ((\\langle v_1, x \\rangle, ..., \\langle v_m, x \\rangle, x) \\in \\mathbb{R}^{m + d} : x \\in \\text{set} \\}.``
+"""
+struct SetWithDotProducts{S,V} <: AbstractVectorSet
+    set::S
+    vectors::Vector{V}
+end
+
+function dimension(s::SetWithDotProducts)
+    return length(s.vectors) + dimension(s.set)
+end
+
+function dual_set(s::SetWithDotProducts)
+    return LinearCombinationInSet(s.set, s.vectors)
+end
+
+function dual_set_type(::Type{SetWithDotProducts{S,V}}) where {S,V}
+    return LinearCombinationInSet{S,V}
+end
+
+"""
+    LinearCombinationInSet{S,V}(set::S, matrices::Vector{V})
+
+Given a set `set` of dimension `d` and `m` vectors `v_1`, ..., `v_m` given in `vectors`, this is the set:
+``\\{ ((y, x) \\in \\mathbb{R}^{m + d} : \\sum_{i=1}^m y_i v_i + x \\in \\text{set} \\}.``
+"""
+struct LinearCombinationInSet{S,V} <: AbstractVectorSet
+    set::S
+    vectors::Vector{V}
+end
+
+dimension(s::LinearCombinationInSet) = length(s.vectors) + simension(s.set)
+
+function dual_set(s::LinearCombinationInSet)
+    return SetWithDotProducts(s.side_dimension, s.matrices)
+end
+
+function dual_set_type(::Type{LinearCombinationInSet{S,V}}) where {S,V}
+    return SetWithDotProducts{S,V}
+end
+
 """
     SOS1{T<:Real}(weights::Vector{T})