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add nuclear norm and spectral norm cone sets (for general/nonsymmetric matrices) #976
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66ac26c
define nuclear and spectral norm cones in sets.jl
chriscoey 10bda57
add to apireference
chriscoey 505204a
add nuclear and spectral tests
chriscoey 87d677f
add Utilities tests and fix some apparent bugs in existing tests
chriscoey fe2aa25
clarify new cone definitions
chriscoey b83290f
start on bridge file
chriscoey 592a75c
add NormSpectralBridge; modify cone definition
chriscoey 393d761
add spectral bridge and start on bridge tests
chriscoey 1c1bfd7
get bridges tests working for spectral bridge
chriscoey 83f4e87
get nuclear bridge working
chriscoey 89a23b9
use better conic tests for spectral and nuclear cones
chriscoey 04e4c91
address comments
chriscoey cae2e41
try allowing rows >= columns
chriscoey e81356a
remove restriction on dimensions
chriscoey 6498687
Merge branch 'master' into addspectralnormcone
chriscoey ead2edd
add spec/nuc cones to MOF
chriscoey 395583b
add primalstart for spectral and dualstart for nuclear
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,199 @@ | ||
| """ | ||
| NormSpectralBridge{T} | ||
|
|
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| The `NormSpectralCone` is representable with a PSD constraint, since | ||
| ``t \\ge \\sigma_1(X)`` if and only if ``[tI X^\\top; X tI] \\succ 0``. | ||
| """ | ||
| struct NormSpectralBridge{T, F, G} <: AbstractBridge | ||
| row_dim::Int # row dimension of X | ||
| column_dim::Int # column dimension of X | ||
| psd_index::CI{F, MOI.PositiveSemidefiniteConeTriangle} | ||
| end | ||
| function bridge_constraint(::Type{NormSpectralBridge{T, F, G}}, model::MOI.ModelLike, f::G, s::MOI.NormSpectralCone) where {T, F, G} | ||
| f_scalars = MOIU.eachscalar(f) | ||
| t = f_scalars[1] | ||
| row_dim = s.row_dim | ||
| column_dim = s.column_dim | ||
| side_dim = row_dim + column_dim | ||
| psd_set = MOI.PositiveSemidefiniteConeTriangle(side_dim) | ||
| psd_func = MOIU.zero_with_output_dimension(F, MOI.dimension(psd_set)) | ||
| for i in 1:side_dim | ||
| MOIU.operate_output_index!(+, T, trimap(i, i), psd_func, t) | ||
| end | ||
| X_idx = 2 | ||
| for j in 1:column_dim, i in (column_dim + 1):side_dim | ||
| MOIU.operate_output_index!(+, T, trimap(i, j), psd_func, f_scalars[X_idx]) | ||
| X_idx += 1 | ||
| end | ||
| psd_index = MOI.add_constraint(model, psd_func, psd_set) | ||
| return NormSpectralBridge{T, F, G}(row_dim, column_dim, psd_index) | ||
| end | ||
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| MOI.supports_constraint(::Type{NormSpectralBridge{T}}, ::Type{<:MOI.AbstractVectorFunction}, ::Type{MOI.NormSpectralCone}) where T = true | ||
| MOIB.added_constrained_variable_types(::Type{<:NormSpectralBridge}) = Tuple{DataType}[] | ||
| MOIB.added_constraint_types(::Type{NormSpectralBridge{T, F, G}}) where {T, F, G} = [(F, MOI.PositiveSemidefiniteConeTriangle)] | ||
| function concrete_bridge_type(::Type{<:NormSpectralBridge{T}}, G::Type{<:MOI.AbstractVectorFunction}, ::Type{MOI.NormSpectralCone}) where T | ||
| F = MOIU.promote_operation(vcat, T, MOIU.scalar_type(G), T) | ||
| return NormSpectralBridge{T, F, G} | ||
| end | ||
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| # Attributes, Bridge acting as a model | ||
| MOI.get(bridge::NormSpectralBridge{T, F, G}, ::MOI.NumberOfConstraints{F, MOI.PositiveSemidefiniteConeTriangle}) where {T, F, G} = 1 | ||
| MOI.get(bridge::NormSpectralBridge{T, F, G}, ::MOI.ListOfConstraintIndices{F, MOI.PositiveSemidefiniteConeTriangle}) where {T, F, G} = [bridge.psd_index] | ||
|
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| # References | ||
| MOI.delete(model::MOI.ModelLike, bridge::NormSpectralBridge) = MOI.delete(model, bridge.psd_index) | ||
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| # Attributes, Bridge acting as a constraint | ||
| function MOI.get(model::MOI.ModelLike, ::MOI.ConstraintFunction, bridge::NormSpectralBridge{T, F, G}) where {T, F, G} | ||
| psd_func = MOIU.eachscalar(MOI.get(model, MOI.ConstraintFunction(), bridge.psd_index)) | ||
| t = psd_func[1] | ||
| side_dim = bridge.row_dim + bridge.column_dim | ||
| X = psd_func[[trimap(i, j) for j in 1:bridge.column_dim for i in (bridge.column_dim + 1):side_dim]] | ||
| return MOIU.convert_approx(G, MOIU.operate(vcat, T, t, X)) | ||
| end | ||
| MOI.get(model::MOI.ModelLike, ::MOI.ConstraintSet, bridge::NormSpectralBridge) = MOI.NormSpectralCone(bridge.row_dim, bridge.column_dim) | ||
| MOI.supports(::MOI.ModelLike, ::MOI.ConstraintPrimalStart, ::Type{<:NormSpectralBridge}) = true | ||
| function MOI.get(model::MOI.ModelLike, attr::Union{MOI.ConstraintPrimal, MOI.ConstraintPrimalStart}, bridge::NormSpectralBridge) | ||
| primal = MOI.get(model, attr, bridge.psd_index) | ||
| t = primal[1] | ||
| side_dim = bridge.row_dim + bridge.column_dim | ||
| X = primal[[trimap(i, j) for j in 1:bridge.column_dim for i in (bridge.column_dim + 1):side_dim]] | ||
| return vcat(t, X) | ||
| end | ||
| function MOI.set(model::MOI.ModelLike, ::MOI.ConstraintPrimalStart, bridge::NormSpectralBridge{T}, value) where T | ||
| column_dim = bridge.column_dim | ||
| side_dim = bridge.row_dim + column_dim | ||
| primal = zeros(T, div(side_dim * (side_dim + 1), 2)) | ||
| for i in 1:side_dim | ||
| primal[trimap(i, i)] = value[1] | ||
| end | ||
| X_idx = 2 | ||
| for j in 1:column_dim, i in (column_dim + 1):side_dim | ||
| primal[trimap(i, j)] = value[X_idx] | ||
| X_idx += 1 | ||
| end | ||
| MOI.set(model, MOI.ConstraintPrimalStart(), bridge.psd_index, primal) | ||
| return | ||
| end | ||
| # Given [U X'; X V] is dual on PSD constraint, the dual on NormSpectralCone | ||
| # constraint is (tr(U) + tr(V), 2X) in NormNuclearCone. | ||
| function MOI.get(model::MOI.ModelLike, ::MOI.ConstraintDual, bridge::NormSpectralBridge) | ||
| dual = MOI.get(model, MOI.ConstraintDual(), bridge.psd_index) | ||
| column_dim = bridge.column_dim | ||
| side_dim = bridge.row_dim + column_dim | ||
| t = sum(dual[trimap(i, i)] for i in 1:side_dim) | ||
| X = 2 * dual[[trimap(i, j) for j in 1:column_dim for i in (column_dim + 1):side_dim]] | ||
| return vcat(t, X) | ||
| end | ||
|
|
||
| """ | ||
| NormNuclearBridge{T} | ||
|
|
||
| The `NormNuclearCone` is representable with an SDP constraint and extra variables, | ||
| since ``t \\ge \\sum_i \\sigma_i (X)`` if and only if there exists symmetric | ||
| matrices ``U, V`` such that ``[U X^\\top; X V] \\succ 0`` and ``t \\ge (tr(U) + tr(V)) / 2``. | ||
| """ | ||
| struct NormNuclearBridge{T, F, G, H} <: AbstractBridge | ||
| row_dim::Int # row dimension of X | ||
| column_dim::Int # column dimension of X | ||
| U::Vector{MOI.VariableIndex} | ||
| V::Vector{MOI.VariableIndex} | ||
| ge_index::CI{F, MOI.GreaterThan{T}} | ||
| psd_index::CI{G, MOI.PositiveSemidefiniteConeTriangle} | ||
| end | ||
| function bridge_constraint(::Type{NormNuclearBridge{T, F, G, H}}, model::MOI.ModelLike, f::H, s::MOI.NormNuclearCone) where {T, F, G, H} | ||
| f_scalars = MOIU.eachscalar(f) | ||
| row_dim = s.row_dim | ||
| column_dim = s.column_dim | ||
| side_dim = row_dim + column_dim | ||
| U_dim = div(column_dim * (column_dim + 1), 2) | ||
| V_dim = div(row_dim * (row_dim + 1), 2) | ||
| U = MOI.add_variables(model, U_dim) | ||
| V = MOI.add_variables(model, V_dim) | ||
| diag_vars = vcat([U[trimap(i, i)] for i in 1:column_dim], [V[trimap(i, i)] for i in 1:row_dim]) | ||
| ge_index = MOIU.normalize_and_add_constraint(model, MOIU.operate(-, T, f_scalars[1], MOIU.operate!(/, T, MOIU.operate(sum, T, diag_vars), T(2))), MOI.GreaterThan(zero(T)), allow_modify_function=true) | ||
| psd_set = MOI.PositiveSemidefiniteConeTriangle(side_dim) | ||
| psd_func = MOI.VectorOfVariables(U) | ||
| nuc_dim = 1 + row_dim * column_dim | ||
| for i in 1:row_dim | ||
| row_i = (1 + i):row_dim:nuc_dim | ||
| psd_func = MOIU.operate(vcat, T, psd_func, f_scalars[row_i]) | ||
| psd_func = MOIU.operate(vcat, T, psd_func, MOI.VectorOfVariables(V[[trimap(i, j) for j in 1:i]])) | ||
| end | ||
| psd_index = MOI.add_constraint(model, psd_func, psd_set) | ||
| return NormNuclearBridge{T, F, G, H}(row_dim, column_dim, U, V, ge_index, psd_index) | ||
| end | ||
|
|
||
| MOI.supports_constraint(::Type{NormNuclearBridge{T}}, ::Type{<:MOI.AbstractVectorFunction}, ::Type{MOI.NormNuclearCone}) where T = true | ||
| MOIB.added_constrained_variable_types(::Type{<:NormNuclearBridge}) = Tuple{DataType}[] | ||
| MOIB.added_constraint_types(::Type{NormNuclearBridge{T, F, G, H}}) where {T, F, G, H} = [(F, MOI.GreaterThan{T}), (G, MOI.PositiveSemidefiniteConeTriangle)] | ||
| function concrete_bridge_type(::Type{<:NormNuclearBridge{T}}, H::Type{<:MOI.AbstractVectorFunction}, ::Type{MOI.NormNuclearCone}) where T | ||
| S = MOIU.scalar_type(H) | ||
| F = MOIU.promote_operation(-, T, S, MOIU.promote_operation(/, T, MOIU.promote_operation(+, T, T, T), T)) | ||
| G = MOIU.promote_operation(vcat, T, MOI.VectorOfVariables, H) | ||
| return NormNuclearBridge{T, F, G, H} | ||
| end | ||
|
|
||
| # Attributes, Bridge acting as a model | ||
| MOI.get(bridge::NormNuclearBridge, ::MOI.NumberOfVariables) = length(bridge.U) + length(bridge.V) | ||
| MOI.get(bridge::NormNuclearBridge, ::MOI.ListOfVariableIndices) = vcat(bridge.U, bridge.V) | ||
| MOI.get(bridge::NormNuclearBridge{T, F, G, H}, ::MOI.NumberOfConstraints{F, MOI.GreaterThan{T}}) where {T, F, G, H} = 1 | ||
| MOI.get(bridge::NormNuclearBridge{T, F, G, H}, ::MOI.NumberOfConstraints{G, MOI.PositiveSemidefiniteConeTriangle}) where {T, F, G, H} = 1 | ||
| MOI.get(bridge::NormNuclearBridge{T, F, G, H}, ::MOI.ListOfConstraintIndices{F, MOI.GreaterThan{T}}) where {T, F, G, H} = [bridge.ge_index] | ||
| MOI.get(bridge::NormNuclearBridge{T, F, G, H}, ::MOI.ListOfConstraintIndices{G, MOI.PositiveSemidefiniteConeTriangle}) where {T, F, G, H} = [bridge.psd_index] | ||
|
|
||
| # References | ||
| function MOI.delete(model::MOI.ModelLike, bridge::NormNuclearBridge) | ||
| MOI.delete(model, bridge.ge_index) | ||
| MOI.delete(model, bridge.psd_index) | ||
| MOI.delete(model, bridge.U) | ||
| MOI.delete(model, bridge.V) | ||
| end | ||
|
|
||
| # Attributes, Bridge acting as a constraint | ||
| function MOI.get(model::MOI.ModelLike, ::MOI.ConstraintFunction, bridge::NormNuclearBridge{T, F, G, H}) where {T, F, G, H} | ||
| ge_func = MOI.get(model, MOI.ConstraintFunction(), bridge.ge_index) | ||
| psd_func = MOIU.eachscalar(MOI.get(model, MOI.ConstraintFunction(), bridge.psd_index)) | ||
| column_dim = bridge.column_dim | ||
| side_dim = bridge.row_dim + column_dim | ||
| t = MOIU.operate(+, T, ge_func, MOIU.operate(/, T, MOIU.operate(+, T, [psd_func[trimap(i, i)] for i in 1:side_dim]...), T(2))) | ||
| t = MOIU.remove_variable(MOIU.remove_variable(t, bridge.U), bridge.V) | ||
| X = psd_func[[trimap(i, j) for j in 1:bridge.column_dim for i in (bridge.column_dim + 1):side_dim]] | ||
| return MOIU.convert_approx(H, MOIU.operate(vcat, T, t, X)) | ||
| end | ||
| MOI.get(model::MOI.ModelLike, ::MOI.ConstraintSet, bridge::NormNuclearBridge) = MOI.NormNuclearCone(bridge.row_dim, bridge.column_dim) | ||
| MOI.supports(::MOI.ModelLike, ::MOI.ConstraintDualStart, ::Type{<:NormNuclearBridge}) = true | ||
| function MOI.get(model::MOI.ModelLike, ::MOI.ConstraintPrimal, bridge::NormNuclearBridge) | ||
| ge_primal = MOI.get(model, MOI.ConstraintPrimal(), bridge.ge_index) | ||
| psd_primal = MOI.get(model, MOI.ConstraintPrimal(), bridge.psd_index) | ||
| side_dim = bridge.row_dim + bridge.column_dim | ||
| t = ge_primal + sum(psd_primal[trimap(i, i)] for i in 1:side_dim) / 2 | ||
| X = psd_primal[[trimap(i, j) for j in 1:bridge.column_dim for i in (bridge.column_dim + 1):side_dim]] | ||
| return vcat(t, X) | ||
| end | ||
| # Given t is dual on GreaterThan constraint and [U X'; X V] is dual on PSD constraint, | ||
| # the dual on NormNuclearCone constraint is (t, 2X) in NormNuclearCone. | ||
| function MOI.get(model::MOI.ModelLike, attr::Union{MOI.ConstraintDual, MOI.ConstraintDualStart}, bridge::NormNuclearBridge) | ||
| t = MOI.get(model, attr, bridge.ge_index) | ||
| psd_dual = MOI.get(model, attr, bridge.psd_index) | ||
| side_dim = bridge.row_dim + bridge.column_dim | ||
| X = 2 * psd_dual[[trimap(i, j) for j in 1:bridge.column_dim for i in (bridge.column_dim + 1):side_dim]] | ||
| return vcat(t, X) | ||
| end | ||
| function MOI.set(model::MOI.ModelLike, ::MOI.ConstraintDualStart, bridge::NormNuclearBridge{T}, value) where T | ||
| MOI.set(model, MOI.ConstraintDualStart(), bridge.ge_index, value[1]) | ||
| column_dim = bridge.column_dim | ||
| side_dim = bridge.row_dim + column_dim | ||
| dual = zeros(T, div(side_dim * (side_dim + 1), 2)) | ||
| for i in 1:side_dim | ||
| dual[trimap(i, i)] = value[1] | ||
| end | ||
| X_idx = 2 | ||
| for j in 1:column_dim, i in (column_dim + 1):side_dim | ||
| dual[trimap(i, j)] = value[X_idx] / 2 | ||
| X_idx += 1 | ||
| end | ||
| MOI.set(model, MOI.ConstraintDualStart(), bridge.psd_index, dual) | ||
| return | ||
| end | ||
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For ConstraintDualStart, I would set all
trimap(i, i)to the dual oftdivided by2side_dim. It might not be the only possibility but it works.There was a problem hiding this comment.
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OK should I make another PR for that?
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I'm wrong, that won't make the matrix PSD.
In fact, it's easier to discuss the ConstraintPrimalStart of NormNuclearBridge.
So suppose you have X and t set.
[A X'; X B] >= 0 is equivalent to A >= X' inv(B) X. If X = U S V', then, it's equivalent to
V' * A * V <= S * U' * inv(B) * U * S
then, by choosing A = V * S * V' and B = U * S * U', you get
S <= S * inv(S) * S which is satisfied.
As the trace must sum up to t, you set A = V * S * V' * (t / tr(S)) and B = U * S * U' * (t / tr(S).
The matrix [A X'; X B] >= will be PSD iff t >= sum sigma_i(X) so it seems like a good candidate for starting value :)
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if that works for constraint primal start of NormNuclear then should it also work for constraint dual start of NormSpectral?
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Yes, it was just easier to reason in the primal than in the dual ^^
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OK rather than having you explain the details to me, it's probably easier if you implement those new start functions