diff --git a/src/Dualization.jl b/src/Dualization.jl
index 5f3e0e0d..bca8b5fb 100644
--- a/src/Dualization.jl
+++ b/src/Dualization.jl
@@ -22,6 +22,7 @@ include("dualize.jl")
 include("MOI_wrapper.jl")
 include("vectorize_emulator.jl")
 include("attributes.jl")
+include("supports.jl")
 
 export dualize
 export dual_optimizer
diff --git a/src/MOI_wrapper.jl b/src/MOI_wrapper.jl
index b1e0bdb4..977e53b7 100644
--- a/src/MOI_wrapper.jl
+++ b/src/MOI_wrapper.jl
@@ -101,56 +101,6 @@ end
 
 DualOptimizer() = error("DualOptimizer must have a solver attached")
 
-MOI.supports(::DualOptimizer, ::MOI.ObjectiveSense) = true
-
-function MOI.supports(
-    optimizer::DualOptimizer{T},
-    ::MOI.ObjectiveFunction{F},
-) where {T,F}
-    # If the objective function is `MOI.VariableIndex` or
-    # `MOI.ScalarAffineFunction`, then a `MOI.ScalarAffineFunction` is set as
-    # the objective function for the dual problem.
-    # If it is `MOI.ScalarQuadraticFunction` , a `MOI.ScalarQuadraticFunction`
-    # is set as objective function for the dual problem.
-    attr = if F <: MOI.ScalarQuadraticFunction
-        MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{T}}()
-    else
-        MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}()
-    end
-    return supported_objective(F) &&
-           MOI.supports(optimizer.dual_problem.dual_model, attr)
-end
-
-function MOI.supports_constraint(
-    optimizer::DualOptimizer{T},
-    ::Type{<:Union{MOI.VariableIndex,MOI.ScalarAffineFunction{T}}},
-    S::Type{<:MOI.AbstractScalarSet},
-) where {T}
-    D = _dual_set_type(S)
-    if D === nothing
-        return false
-    end
-    model = optimizer.dual_problem.dual_model
-    if D <: MOI.AbstractVectorSet # The dual of `EqualTo` is `Reals`
-        return MOI.supports_add_constrained_variables(model, D)
-    else
-        return MOI.supports_add_constrained_variable(model, D)
-    end
-end
-
-function MOI.supports_constraint(
-    optimizer::DualOptimizer{T},
-    ::Type{<:Union{MOI.VectorOfVariables,MOI.VectorAffineFunction{T}}},
-    S::Type{<:MOI.AbstractVectorSet},
-) where {T}
-    D = _dual_set_type(S)
-    if D === nothing
-        return false
-    end
-    model = optimizer.dual_problem.dual_model
-    return MOI.supports_add_constrained_variables(model, D)
-end
-
 function _change_constant(
     model,
     ci::MOI.ConstraintIndex{<:MOI.ScalarAffineFunction,S},
@@ -205,34 +155,6 @@ function MOI.modify(
     return
 end
 
-function MOI.supports_add_constrained_variables(
-    optimizer::DualOptimizer{T},
-    ::Type{MOI.Reals},
-) where {T}
-    return MOI.supports_constraint(
-        optimizer.dual_problem.dual_model,
-        MOI.ScalarAffineFunction{T},
-        MOI.EqualTo{T},
-    )
-    # If `_dual_set_type(MOI.Reals)` was `MOI.Zeros`,
-    # we would not need this method as special case of the one below
-end
-
-function MOI.supports_add_constrained_variables(
-    optimizer::DualOptimizer{T},
-    S::Type{<:MOI.AbstractVectorSet},
-) where {T}
-    D = _dual_set_type(S)
-    if D === nothing
-        return false
-    end
-    return MOI.supports_constraint(
-        optimizer.dual_problem.dual_model,
-        MOI.VectorAffineFunction{T},
-        D,
-    )
-end
-
 function MOI.copy_to(dest::DualOptimizer, src::MOI.ModelLike)
     MOI.empty!(dest)
     dualize(
diff --git a/src/supports.jl b/src/supports.jl
new file mode 100644
index 00000000..58196ae6
--- /dev/null
+++ b/src/supports.jl
@@ -0,0 +1,146 @@
+MOI.supports(::DualOptimizer, ::MOI.ObjectiveSense) = true
+
+function MOI.supports(
+    optimizer::DualOptimizer{T},
+    ::MOI.ObjectiveFunction{F},
+) where {T,F}
+    # If the objective function is `MOI.VariableIndex` or
+    # `MOI.ScalarAffineFunction`, then a `MOI.ScalarAffineFunction` is set as
+    # the objective function for the dual problem.
+    # If it is `MOI.ScalarQuadraticFunction` , a `MOI.ScalarQuadraticFunction`
+    # is set as objective function for the dual problem.
+    attr = if F <: MOI.ScalarQuadraticFunction
+        MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{T}}()
+    else
+        MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}()
+    end
+    return supported_objective(F) &&
+           MOI.supports(optimizer.dual_problem.dual_model, attr)
+end
+
+# Generic fallback for `VariableBridgingCost`/`ConstraintBridgingCost`
+# calls `dual_attribute` which won't work for sets that don't implement `dual_set_type`
+# so we define custom `get` methods instead.
+
+function MOI.get(
+    optimizer::DualOptimizer,
+    attr::Union{MOI.VariableBridgingCost,MOI.ConstraintBridgingCost},
+)
+    return MOI.get_fallback(optimizer, attr)
+end
+
+function MOI.get(
+    optimizer::DualOptimizer{T},
+    ::MOI.ConstraintBridgingCost{
+        <:Union{MOI.VariableIndex,MOI.ScalarAffineFunction{T}},
+        S,
+    },
+) where {T,S<:MOI.AbstractScalarSet}
+    D = _dual_set_type(S)
+    if D === nothing
+        return Inf
+    end
+    return MOI.get(
+        optimizer.dual_problem.dual_model,
+        MOI.VariableBridgingCost{D}(),
+    )
+end
+
+function MOI.supports_constraint(
+    optimizer::DualOptimizer{T},
+    ::Type{<:Union{MOI.VariableIndex,MOI.ScalarAffineFunction{T}}},
+    S::Type{<:MOI.AbstractScalarSet},
+) where {T}
+    D = _dual_set_type(S)
+    if D === nothing
+        return false
+    end
+    model = optimizer.dual_problem.dual_model
+    if D <: MOI.AbstractVectorSet # The dual of `EqualTo` is `Reals`
+        return MOI.supports_add_constrained_variables(model, D)
+    else
+        return MOI.supports_add_constrained_variable(model, D)
+    end
+end
+
+function MOI.get(
+    optimizer::DualOptimizer{T},
+    ::MOI.ConstraintBridgingCost{
+        <:Union{MOI.VectorOfVariables,MOI.VectorAffineFunction{T}},
+        S,
+    },
+) where {T,S<:MOI.AbstractVectorSet}
+    D = _dual_set_type(S)
+    if D === nothing
+        return Inf
+    end
+    return MOI.get(
+        optimizer.dual_problem.dual_model,
+        MOI.VariableBridgingCost{D}(),
+    )
+end
+
+function MOI.supports_constraint(
+    optimizer::DualOptimizer{T},
+    ::Type{<:Union{MOI.VectorOfVariables,MOI.VectorAffineFunction{T}}},
+    S::Type{<:MOI.AbstractVectorSet},
+) where {T}
+    D = _dual_set_type(S)
+    if D === nothing
+        return false
+    end
+    model = optimizer.dual_problem.dual_model
+    return MOI.supports_add_constrained_variables(model, D)
+end
+
+function MOI.get(
+    optimizer::DualOptimizer{T},
+    ::MOI.VariableBridgingCost{MOI.Reals},
+) where {T}
+    return MOI.get(
+        optimizer.dual_problem.dual_model,
+        MOI.ConstraintBridgingCost{MOI.ScalarAffineFunction{T},MOI.EqualTo{T}}(),
+    )
+end
+
+function MOI.supports_add_constrained_variables(
+    optimizer::DualOptimizer{T},
+    ::Type{MOI.Reals},
+) where {T}
+    return MOI.supports_constraint(
+        optimizer.dual_problem.dual_model,
+        MOI.ScalarAffineFunction{T},
+        MOI.EqualTo{T},
+    )
+    # If `_dual_set_type(MOI.Reals)` was `MOI.Zeros`,
+    # we would not need this method as special case of the one below
+end
+
+function MOI.get(
+    optimizer::DualOptimizer{T},
+    ::MOI.VariableBridgingCost{S},
+) where {T,S<:MOI.AbstractVectorSet}
+    D = _dual_set_type(S)
+    if D === nothing
+        return Inf
+    end
+    return MOI.get(
+        optimizer.dual_problem.dual_model,
+        MOI.ConstraintBridgingCost{MOI.VectorAffineFunction{T},D}(),
+    )
+end
+
+function MOI.supports_add_constrained_variables(
+    optimizer::DualOptimizer{T},
+    S::Type{<:MOI.AbstractVectorSet},
+) where {T}
+    D = _dual_set_type(S)
+    if D === nothing
+        return false
+    end
+    return MOI.supports_constraint(
+        optimizer.dual_problem.dual_model,
+        MOI.VectorAffineFunction{T},
+        D,
+    )
+end
diff --git a/test/Solvers/hypatia_test.jl b/test/Solvers/hypatia_test.jl
index 77e343ce..2a3e9bb1 100644
--- a/test/Solvers/hypatia_test.jl
+++ b/test/Solvers/hypatia_test.jl
@@ -5,6 +5,42 @@
 
 import Hypatia
 
+@testset "VariableBridgingCost / ConstraintBridgingCost" begin
+    opt = MOI.instantiate(dual_optimizer(Hypatia.Optimizer))
+    # Specialized `VariableBridgingCost{MOI.Reals}`
+    @test MOI.supports_add_constrained_variables(opt, MOI.Reals)
+    @test MOI.get(opt, MOI.VariableBridgingCost{MOI.Reals}()) == 1
+    # Specialized `VariableBridgingCost{S<:AbstractVectorSet}`, supported branch
+    @test MOI.supports_add_constrained_variables(opt, MOI.Nonnegatives)
+    @test MOI.get(opt, MOI.VariableBridgingCost{MOI.Nonnegatives}()) == 0
+    # Same dispatch, supported with nonzero cost
+    @test MOI.get(opt, MOI.VariableBridgingCost{MOI.Zeros}()) == 1
+    # Same dispatch, `_dual_set_type` returns `nothing`
+    @test !MOI.supports_add_constrained_variables(opt, MOI.SOS1{Float64})
+    @test MOI.get(opt, MOI.VariableBridgingCost{MOI.SOS1{Float64}}()) == Inf
+    # Specialized scalar `ConstraintBridgingCost`, supported branch
+    @test MOI.get(
+        opt,
+        MOI.ConstraintBridgingCost{
+            MOI.ScalarAffineFunction{Float64},
+            MOI.GreaterThan{Float64},
+        }(),
+    ) == 1
+    # Specialized vector `ConstraintBridgingCost`, `_dual_set_type` returns `nothing`
+    @test !MOI.supports_constraint(
+        opt,
+        MOI.VectorOfVariables,
+        MOI.SOS1{Float64},
+    )
+    @test MOI.get(
+        opt,
+        MOI.ConstraintBridgingCost{MOI.VectorOfVariables,MOI.SOS1{Float64}}(),
+    ) == Inf
+    # Generic fallback (no specialized method matches)
+    @test MOI.get(opt, MOI.VariableBridgingCost{MOI.GreaterThan{Float64}}()) ==
+          0
+end
+
 @testset "Solve problems with different coefficient_type" begin
     function mineig(::Type{T}) where {T}
         model = GenericModel{T}(
diff --git a/test/Tests/test_MOI_wrapper.jl b/test/Tests/test_MOI_wrapper.jl
index cdcad6d6..4e8b41d8 100644
--- a/test/Tests/test_MOI_wrapper.jl
+++ b/test/Tests/test_MOI_wrapper.jl
@@ -96,6 +96,10 @@
     @testset "support" begin
         for opt in dual_linear_optimizer
             @test !MOI.supports_constraint(opt, MOI.VariableIndex, MOI.Integer)
+            @test MOI.get(
+                opt,
+                MOI.ConstraintBridgingCost{MOI.VariableIndex,MOI.Integer}(),
+            ) == Inf
             @test MOI.supports(opt, MOI.ObjectiveSense())
         end
         for opt in dual_conic_optimizer
@@ -104,6 +108,13 @@
                 MOI.VectorOfVariables,
                 MOI.PositiveSemidefiniteConeTriangle,
             )
+            @test MOI.get(
+                opt,
+                MOI.ConstraintBridgingCost{
+                    MOI.VectorOfVariables,
+                    MOI.PositiveSemidefiniteConeTriangle,
+                }(),
+            ) == 0
         end
     end