diff --git a/src/Bridges/Bridges.jl b/src/Bridges/Bridges.jl
index 1aeff32883..a05434a576 100644
--- a/src/Bridges/Bridges.jl
+++ b/src/Bridges/Bridges.jl
@@ -284,7 +284,7 @@ julia> MOI.Bridges.runtests(
            end,
        )
 Test Summary:    | Pass  Total  Time
-Bridges.runtests |   32     32  0.8s
+Bridges.runtests |   32     32  0.0s
 ```
 """
 function runtests(args...; kwargs...)
@@ -294,12 +294,100 @@ function runtests(args...; kwargs...)
     return
 end
 
+# This function tests that ConstraintDual is implemented correctly for a bridge
+# by comparing Lagrangian values of the inner and outer models at corresponding
+# primal points. Evaluating the functions also works for nonlinear expressions.
+#
+# For this test to be enabled, the bridge should implement `MOI.supports` and
+# `MOI.set` for `ConstraintDual`. The easiest way to achieve this is with
+# `Union{ConstraintDual,ConstraintDualStart}`.
+function _test_dual(
+    Bridge::Type{<:AbstractBridge},
+    input_fn::Function;
+    dual,
+    eltype,
+    model_eltype,
+)
+    if Bridge <: Objective.AbstractBridge
+        return
+    end
+    inner = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{model_eltype}())
+    mock = MOI.Utilities.MockOptimizer(
+        inner;
+        eval_variable_constraint_dual = false,
+    )
+    model = _bridged_model(Bridge{eltype}, mock)
+    input_fn(model)
+    final_touch(model)
+    final_touch(model)
+    MOI.set(mock, MOI.PrimalStatus(), MOI.FEASIBLE_POINT)
+    # Use an integer point so integer-coefficient vector functions can be
+    # evaluated without inexact conversions. Keep auxiliary primals at zero.
+    primal = 2 * one(model_eltype)
+    x = MOI.get(mock, MOI.ListOfVariableIndices())
+    MOI.set(mock, MOI.VariablePrimal(), x, fill(zero(primal), length(x)))
+    if Bridge <: Variable.AbstractBridge
+        MOI.set(mock, MOI.VariablePrimal(), x, fill(primal, length(x)))
+    else
+        x = MOI.get(model, MOI.ListOfVariableIndices())
+        MOI.set(model, MOI.VariablePrimal(), x, fill(primal, length(x)))
+    end
+    MOI.set(mock, MOI.DualStatus(), MOI.FEASIBLE_POINT)
+    # Some bridges leave auxiliary constraint duals unspecified.
+    for (F, S) in MOI.get(mock, MOI.ListOfConstraintTypesPresent())
+        for ci in MOI.get(mock, MOI.ListOfConstraintIndices{F,S}())
+            set = MOI.get(mock, MOI.ConstraintSet(), ci)
+            MOI.set(
+                mock,
+                MOI.ConstraintDual(),
+                ci,
+                _fake_start(zero(model_eltype), set),
+            )
+        end
+    end
+    for (F, S) in MOI.get(model, MOI.ListOfConstraintTypesPresent())
+        if !MOI.supports(model, MOI.ConstraintDual(), MOI.ConstraintIndex{F,S})
+            return
+        end
+        for ci in MOI.get(model, MOI.ListOfConstraintIndices{F,S}())
+            set = MOI.get(model, MOI.ConstraintSet(), ci)
+            value = _fake_start(dual, set)
+            MOI.set(model, MOI.ConstraintDual(), ci, value)
+            Test.@test MOI.get(model, MOI.ConstraintDual(), ci) ≈ value
+        end
+    end
+    Test.@test isapprox(
+        _test_lagrangian_value(model),
+        _test_lagrangian_value(mock);
+        atol = 1e-6,
+    )
+    return
+end
+
+function _test_lagrangian_value(model::MOI.ModelLike)
+    value = MOI.Utilities.get_fallback(model, MOI.ObjectiveValue())
+    sign = MOI.get(model, MOI.ObjectiveSense()) == MOI.MAX_SENSE ? 1 : -1
+    for (F, S) in MOI.get(model, MOI.ListOfConstraintTypesPresent())
+        for ci in MOI.get(model, MOI.ListOfConstraintIndices{F,S}())
+            f = MOI.get(model, MOI.ConstraintFunction(), ci)
+            primal = MOI.Utilities.eval_variables(model, f) do vi
+                return MOI.get(model, MOI.VariablePrimal(), vi)
+            end
+            set = MOI.get(model, MOI.ConstraintSet(), ci)
+            dual = MOI.get(model, MOI.ConstraintDual(), ci)
+            value += sign * MOI.Utilities._dual_objective_dot(primal, dual, set)
+        end
+    end
+    return value
+end
+
 function _runtests(
     Bridge::Type{<:AbstractBridge},
     input_fn::Function,
     output_fn::Function;
     variable_start = 1.2,
     constraint_start = 1.2,
+    dual = constraint_start,
     eltype = Float64,
     model_eltype = eltype,
     print_inner_model::Bool = false,
@@ -404,6 +492,11 @@ function _runtests(
     Test.@testset "Test delete" begin                                           # COV_EXCL_LINE
         _test_delete(Bridge, model, inner)
     end
+    if !isnothing(dual) && !cannot_unbridge
+        Test.@testset "Test ConstraintDual" begin                               # COV_EXCL_LINE
+            _test_dual(Bridge, input_fn; dual, eltype, model_eltype)
+        end
+    end
     return
 end
 
diff --git a/src/Bridges/Constraint/bridges/AbstractFunctionConversionBridge.jl b/src/Bridges/Constraint/bridges/AbstractFunctionConversionBridge.jl
index e551bd13f6..7de3f09921 100644
--- a/src/Bridges/Constraint/bridges/AbstractFunctionConversionBridge.jl
+++ b/src/Bridges/Constraint/bridges/AbstractFunctionConversionBridge.jl
@@ -281,7 +281,7 @@ function conversion_cost(
     # This is most useful for conic solvers, which often support
     # VectorAffineFunction-in-S, but not VectorOfVariables-in-S.
     #
-    # The value in this function controls the cost of the transformation 
+    # The value in this function controls the cost of the transformation
     # VectorOfVariables-in-S -> VectorAffineFunction-in-S. When we had the
     # previous cost of `1.0`, there might be an alternative bridge
     # VectorOfVariables-in-S -> VectorAffineFunction-in-S2 which also has a cost
diff --git a/src/Bridges/Constraint/bridges/GeoMeanToPowerBridge.jl b/src/Bridges/Constraint/bridges/GeoMeanToPowerBridge.jl
index 191868057a..87aae0a845 100644
--- a/src/Bridges/Constraint/bridges/GeoMeanToPowerBridge.jl
+++ b/src/Bridges/Constraint/bridges/GeoMeanToPowerBridge.jl
@@ -259,7 +259,7 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     ::Type{GeoMeanToPowerBridge{T,F}},
 ) where {T,F}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F,MOI.PowerCone{T}})
@@ -293,7 +293,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::GeoMeanToPowerBridge{T,F},
     start::AbstractVector{T},
 ) where {T,F}
@@ -327,7 +327,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::GeoMeanToPowerBridge,
     ::Nothing,
 )
diff --git a/src/Bridges/Constraint/bridges/GeoMeantoRelEntrBridge.jl b/src/Bridges/Constraint/bridges/GeoMeantoRelEntrBridge.jl
index d0894d6ec9..fb450921dd 100644
--- a/src/Bridges/Constraint/bridges/GeoMeantoRelEntrBridge.jl
+++ b/src/Bridges/Constraint/bridges/GeoMeantoRelEntrBridge.jl
@@ -198,7 +198,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{GeoMeantoRelEntrBridge{T,F,G,H}},
 ) where {T,F,G,H}
     ci_1 = MOI.ConstraintIndex{F,MOI.Nonnegatives}
@@ -277,28 +281,23 @@ end
 # Note log.(w/geomean(w)) = log.(w) .- sum(log, w) / n.
 function MOI.set(
     model::MOI.ModelLike,
-    ::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::GeoMeantoRelEntrBridge,
     value,
 )
     d = length(value) - 1
     u = value[1]
     w = value[2:end]
-    MOI.set(model, MOI.ConstraintDualStart(), bridge.nn_index, [-u])
+    MOI.set(model, attr, bridge.nn_index, [-u])
     relentr_dual =
         vcat(-u / d, value[2:end], u / d * (log.(w) .+ (1 - sum(log, w) / d)))
-    MOI.set(
-        model,
-        MOI.ConstraintDualStart(),
-        bridge.relentr_index,
-        relentr_dual,
-    )
+    MOI.set(model, attr, bridge.relentr_index, relentr_dual)
     return
 end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::GeoMeantoRelEntrBridge,
     ::Nothing,
 )
diff --git a/src/Bridges/Constraint/bridges/IntervalToHyperRectangleBridge.jl b/src/Bridges/Constraint/bridges/IntervalToHyperRectangleBridge.jl
index 97517549b2..85ea0783e8 100644
--- a/src/Bridges/Constraint/bridges/IntervalToHyperRectangleBridge.jl
+++ b/src/Bridges/Constraint/bridges/IntervalToHyperRectangleBridge.jl
@@ -101,7 +101,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{IntervalToHyperRectangleBridge{T,F,G}},
 ) where {T,F,G}
     return MOI.supports(
@@ -157,7 +161,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::IntervalToHyperRectangleBridge,
     value,
 )
diff --git a/src/Bridges/Constraint/bridges/NormNuclearBridge.jl b/src/Bridges/Constraint/bridges/NormNuclearBridge.jl
index e57dfd53e1..68d04eef3e 100644
--- a/src/Bridges/Constraint/bridges/NormNuclearBridge.jl
+++ b/src/Bridges/Constraint/bridges/NormNuclearBridge.jl
@@ -137,7 +137,7 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintPrimalStart,
+    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDual},
     ::Type{NormSpectralBridge{T,F,G}},
 ) where {T,F,G}
     return MOI.supports(
@@ -215,6 +215,28 @@ function MOI.get(
     return vcat(t, X)
 end
 
+function MOI.set(
+    model::MOI.ModelLike,
+    attr::MOI.ConstraintDual,
+    bridge::NormSpectralBridge{T},
+    value::AbstractVector,
+) where {T}
+    column_dim = bridge.column_dim
+    side_dim = bridge.row_dim + column_dim
+    dual = zeros(T, div(side_dim * (side_dim + 1), 2))
+    # Invert the trace and the doubled off-diagonal entries in the getter.
+    for i in 1:side_dim
+        dual[MOI.Utilities.trimap(i, i)] = value[1] / side_dim
+    end
+    k = 2
+    for j in 1:column_dim, i in (column_dim+1):side_dim
+        dual[MOI.Utilities.trimap(i, j)] = value[k] / 2
+        k += 1
+    end
+    MOI.set(model, attr, bridge.psd_index, dual)
+    return
+end
+
 """
     NormNuclearBridge{T,F,G,H} <: Bridges.Constraint.AbstractBridge
 
@@ -444,7 +466,7 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     ::Type{NormNuclearBridge{T,F,G,H}},
 ) where {T,F,G,H}
     ci_1 = MOI.ConstraintIndex{F,MOI.GreaterThan{T}}
@@ -494,11 +516,11 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    ::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::NormNuclearBridge{T},
     value,
 ) where {T}
-    MOI.set(model, MOI.ConstraintDualStart(), bridge.ge_index, value[1])
+    MOI.set(model, attr, 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))
@@ -510,17 +532,17 @@ function MOI.set(
         dual[MOI.Utilities.trimap(i, j)] = value[X_idx] / 2
         X_idx += 1
     end
-    MOI.set(model, MOI.ConstraintDualStart(), bridge.psd_index, dual)
+    MOI.set(model, attr, bridge.psd_index, dual)
     return
 end
 
 function MOI.set(
     model::MOI.ModelLike,
-    ::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::NormNuclearBridge{T},
     ::Nothing,
 ) where {T}
-    MOI.set(model, MOI.ConstraintDualStart(), bridge.ge_index, nothing)
-    MOI.set(model, MOI.ConstraintDualStart(), bridge.psd_index, nothing)
+    MOI.set(model, attr, bridge.ge_index, nothing)
+    MOI.set(model, attr, bridge.psd_index, nothing)
     return
 end
diff --git a/src/Bridges/Constraint/bridges/NormOneBridge.jl b/src/Bridges/Constraint/bridges/NormOneBridge.jl
index e0221d5105..e257805b80 100644
--- a/src/Bridges/Constraint/bridges/NormOneBridge.jl
+++ b/src/Bridges/Constraint/bridges/NormOneBridge.jl
@@ -144,7 +144,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{NormOneBridge{T,F,G}},
 ) where {T,F,G}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F,MOI.Nonnegatives})
@@ -219,7 +223,7 @@ end
 # depending on the sense of `value[1 + i]`.
 function MOI.set(
     model::MOI.ModelLike,
-    ::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::NormOneBridge,
     value,
 )
@@ -233,16 +237,16 @@ function MOI.set(
             nn_dual[1+d+i] = value[1+i]
         end
     end
-    MOI.set(model, MOI.ConstraintDualStart(), bridge.nn_index, nn_dual)
+    MOI.set(model, attr, bridge.nn_index, nn_dual)
     return
 end
 
 function MOI.set(
     model::MOI.ModelLike,
-    ::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::NormOneBridge,
     ::Nothing,
 )
-    MOI.set(model, MOI.ConstraintDualStart(), bridge.nn_index, nothing)
+    MOI.set(model, attr, bridge.nn_index, nothing)
     return
 end
diff --git a/src/Bridges/Constraint/bridges/NumberConversionBridge.jl b/src/Bridges/Constraint/bridges/NumberConversionBridge.jl
index ca1f24fb47..0776fcfa1e 100644
--- a/src/Bridges/Constraint/bridges/NumberConversionBridge.jl
+++ b/src/Bridges/Constraint/bridges/NumberConversionBridge.jl
@@ -123,7 +123,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{NumberConversionBridge{T,F1,S1,F2,S2}},
 ) where {T,F1,S1,F2,S2}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F2,S2})
diff --git a/src/Bridges/Constraint/bridges/RelativeEntropyBridge.jl b/src/Bridges/Constraint/bridges/RelativeEntropyBridge.jl
index 2f5519475f..7df2247cc8 100644
--- a/src/Bridges/Constraint/bridges/RelativeEntropyBridge.jl
+++ b/src/Bridges/Constraint/bridges/RelativeEntropyBridge.jl
@@ -185,7 +185,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{RelativeEntropyBridge{T,F,G,H}},
 ) where {T,F,G,H}
     ci_1 = MOI.ConstraintIndex{F,MOI.GreaterThan{T}}
@@ -275,7 +279,7 @@ end
 # r_i = exp(W(-w_i / (-v_i * e))) * (-v_i * e), but this is more complicated.
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::RelativeEntropyBridge,
     value,
 )
@@ -289,7 +293,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::RelativeEntropyBridge,
     ::Nothing,
 )
diff --git a/src/Bridges/Constraint/bridges/SOCtoPSDBridge.jl b/src/Bridges/Constraint/bridges/SOCtoPSDBridge.jl
index 0d483a9797..1ce6ffaf6b 100644
--- a/src/Bridges/Constraint/bridges/SOCtoPSDBridge.jl
+++ b/src/Bridges/Constraint/bridges/SOCtoPSDBridge.jl
@@ -218,7 +218,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{<:RSOCtoPSDBridge{T,F,G}},
 ) where {T,F,G}
     return MOI.supports(
diff --git a/src/Bridges/Constraint/bridges/ScalarSlackBridge.jl b/src/Bridges/Constraint/bridges/ScalarSlackBridge.jl
index da0a3fdb3e..c5445e4c15 100644
--- a/src/Bridges/Constraint/bridges/ScalarSlackBridge.jl
+++ b/src/Bridges/Constraint/bridges/ScalarSlackBridge.jl
@@ -80,7 +80,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{<:_AbstractSlackBridge{T,VF,ZS,F,S}},
 ) where {T,VF,ZS,F,S}
     ret = true
@@ -152,7 +156,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::_AbstractSlackBridge{T},
     value,
 ) where {T}
diff --git a/src/Bridges/Constraint/bridges/ScalarizeBridge.jl b/src/Bridges/Constraint/bridges/ScalarizeBridge.jl
index 0038cb4553..ea4074a35d 100644
--- a/src/Bridges/Constraint/bridges/ScalarizeBridge.jl
+++ b/src/Bridges/Constraint/bridges/ScalarizeBridge.jl
@@ -153,7 +153,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{ScalarizeBridge{T,F,S}},
 ) where {T,F,S}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F,S})
@@ -211,7 +215,7 @@ end
 
 function MOI.get(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::ScalarizeBridge,
 )
     values = MOI.get.(model, attr, bridge.scalar_constraints)
@@ -223,7 +227,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::ScalarizeBridge,
     value,
 )
diff --git a/src/Bridges/Constraint/bridges/SplitComplexEqualToBridge.jl b/src/Bridges/Constraint/bridges/SplitComplexEqualToBridge.jl
index ebebe8e8cb..e330031adf 100644
--- a/src/Bridges/Constraint/bridges/SplitComplexEqualToBridge.jl
+++ b/src/Bridges/Constraint/bridges/SplitComplexEqualToBridge.jl
@@ -160,7 +160,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{<:SplitComplexEqualToBridge{T,F}},
 ) where {T,F}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F,MOI.EqualTo{T}})
@@ -194,7 +198,11 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     bridge::SplitComplexEqualToBridge{T},
     value,
 ) where {T}
@@ -209,7 +217,11 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     bridge::SplitComplexEqualToBridge,
     ::Nothing,
 )
diff --git a/src/Bridges/Constraint/bridges/SplitComplexZerosBridge.jl b/src/Bridges/Constraint/bridges/SplitComplexZerosBridge.jl
index 3e12f77136..0a626c2c81 100644
--- a/src/Bridges/Constraint/bridges/SplitComplexZerosBridge.jl
+++ b/src/Bridges/Constraint/bridges/SplitComplexZerosBridge.jl
@@ -158,7 +158,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{<:SplitComplexZerosBridge{T,F}},
 ) where {T,F}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F,MOI.Zeros})
@@ -190,7 +194,11 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     bridge::SplitComplexZerosBridge{T},
     value,
 ) where {T}
@@ -210,7 +218,11 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     bridge::SplitComplexZerosBridge{T},
     ::Nothing,
 ) where {T}
diff --git a/src/Bridges/Constraint/bridges/SplitHyperRectangleBridge.jl b/src/Bridges/Constraint/bridges/SplitHyperRectangleBridge.jl
index 5309b56f08..ebfb728d69 100644
--- a/src/Bridges/Constraint/bridges/SplitHyperRectangleBridge.jl
+++ b/src/Bridges/Constraint/bridges/SplitHyperRectangleBridge.jl
@@ -196,7 +196,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{<:SplitHyperRectangleBridge{T,G}},
 ) where {T,G}
     return MOI.supports(model, attr, MOI.ConstraintIndex{G,MOI.Nonnegatives})
@@ -280,7 +284,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::SplitHyperRectangleBridge{T},
     ::Nothing,
 ) where {T}
@@ -293,7 +297,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::SplitHyperRectangleBridge{T},
     value::AbstractVector{T},
 ) where {T}
diff --git a/src/Bridges/Constraint/bridges/SplitIntervalBridge.jl b/src/Bridges/Constraint/bridges/SplitIntervalBridge.jl
index af393a1863..41e01b7ab0 100644
--- a/src/Bridges/Constraint/bridges/SplitIntervalBridge.jl
+++ b/src/Bridges/Constraint/bridges/SplitIntervalBridge.jl
@@ -211,7 +211,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{SplitIntervalBridge{T,F,S,LS,US}},
 ) where {T,F,S,LS,US}
     ci_1 = MOI.ConstraintIndex{F,LS}
@@ -296,7 +300,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::SplitIntervalBridge{T},
     value,
 ) where {T}
diff --git a/src/Bridges/Constraint/bridges/SquareBridge.jl b/src/Bridges/Constraint/bridges/SquareBridge.jl
index fe3132a791..1f7302444e 100644
--- a/src/Bridges/Constraint/bridges/SquareBridge.jl
+++ b/src/Bridges/Constraint/bridges/SquareBridge.jl
@@ -279,7 +279,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{SquareBridge{T,F,G,TT,ST}},
 ) where {T,F,G,TT,ST}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F,TT}) &
@@ -417,7 +421,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::SquareBridge,
     value,
 )
@@ -453,7 +457,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::SquareBridge,
     ::Nothing,
 )
diff --git a/src/Bridges/Constraint/bridges/VectorizeBridge.jl b/src/Bridges/Constraint/bridges/VectorizeBridge.jl
index 50a1ea4854..26bda2be6e 100644
--- a/src/Bridges/Constraint/bridges/VectorizeBridge.jl
+++ b/src/Bridges/Constraint/bridges/VectorizeBridge.jl
@@ -108,7 +108,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{VectorizeBridge{T,F,S,G}},
 ) where {T,F,S,G}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F,S})
@@ -183,7 +187,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::VectorizeBridge,
     value,
 )
diff --git a/src/Bridges/Constraint/set_map.jl b/src/Bridges/Constraint/set_map.jl
index 70062db687..450c99f3d2 100644
--- a/src/Bridges/Constraint/set_map.jl
+++ b/src/Bridges/Constraint/set_map.jl
@@ -24,7 +24,7 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart,MOI.ConstraintDual},
     ::Type{<:BridgeType{T,F2,F3}},
 ) where {T,F2,F3}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F2,S2}) ||
@@ -205,7 +205,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::BT,
     value,
 ) where {BT<:MultiSetMapBridge}
@@ -263,7 +263,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{<:SetMapBridge{T,S2,S1,F}},
 ) where {T,S2,S1,F}
     return MOI.supports(model, attr, MOI.ConstraintIndex{F,S2})
diff --git a/src/Bridges/Variable/set_map.jl b/src/Bridges/Variable/set_map.jl
index 21c9af8105..2b3a53ad8e 100644
--- a/src/Bridges/Variable/set_map.jl
+++ b/src/Bridges/Variable/set_map.jl
@@ -155,7 +155,7 @@ end
 
 function MOI.set(
     model::MOI.ModelLike,
-    attr::MOI.ConstraintDualStart,
+    attr::Union{MOI.ConstraintDualStart,MOI.ConstraintDual},
     bridge::SetMapBridge,
     value,
 )
@@ -272,7 +272,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{<:SetMapBridge{T,S1}},
 ) where {T,S1<:MOI.AbstractScalarSet}
     return MOI.supports(model, attr, MOI.ConstraintIndex{MOI.VariableIndex,S1})
@@ -351,7 +355,11 @@ end
 
 function MOI.supports(
     model::MOI.ModelLike,
-    attr::Union{MOI.ConstraintPrimalStart,MOI.ConstraintDualStart},
+    attr::Union{
+        MOI.ConstraintPrimalStart,
+        MOI.ConstraintDualStart,
+        MOI.ConstraintDual,
+    },
     ::Type{<:SetMapBridge{T,S1}},
 ) where {T,S1<:MOI.AbstractVectorSet}
     CI = MOI.ConstraintIndex{MOI.VectorOfVariables,S1}
diff --git a/src/Utilities/results.jl b/src/Utilities/results.jl
index d1f199334d..b7ba290d6e 100644
--- a/src/Utilities/results.jl
+++ b/src/Utilities/results.jl
@@ -122,6 +122,8 @@ _variable_set_in_dual_objective(::Type{<:MOI.AbstractSet}) = false
 
 _variable_set_in_dual_objective(::Type{<:MOI.EqualTo}) = true
 
+_variable_set_in_dual_objective(::Type{<:MOI.Parameter}) = true
+
 _variable_set_in_dual_objective(::Type{<:MOI.GreaterThan}) = true
 
 _variable_set_in_dual_objective(::Type{<:MOI.LessThan}) = true
@@ -130,9 +132,11 @@ _variable_set_in_dual_objective(::Type{<:MOI.Interval}) = true
 
 _variable_set_in_dual_objective(::Type{<:MOI.HyperRectangle}) = true
 
-_dual_objective_dot(x, y, set) = set_dot(x, y, set)
+_dual_objective_dot(x, y, set) = real(set_dot(x, y, set))
 
-_dual_objective_dot(x, y, set::MOI.EqualTo) = (x - set.value) * y
+function _dual_objective_dot(x, y, set::Union{MOI.EqualTo,MOI.Parameter})
+    return real(set_dot(x - set.value, y, set))
+end
 
 _dual_objective_dot(x, y, set::MOI.LessThan) = (x - set.upper) * y
 
diff --git a/src/Utilities/set_dot.jl b/src/Utilities/set_dot.jl
index 5abbdc7641..32d04e3156 100644
--- a/src/Utilities/set_dot.jl
+++ b/src/Utilities/set_dot.jl
@@ -39,9 +39,9 @@ function triangle_dot(
         for j in 1:i
             k += 1
             if i == j
-                result = MA.add_mul!!(result, x[k], y[k])
+                result = MA.add_mul!!(result, conj(x[k]), y[k])
             else
-                result = MA.add_mul!!(result, 2, x[k], y[k])
+                result = MA.add_mul!!(result, 2, conj(x[k]), y[k])
             end
         end
     end
diff --git a/test/Bridges/Constraint/test_ScalarFunctionizeBridge.jl b/test/Bridges/Constraint/test_ScalarFunctionizeBridge.jl
index eec295bef8..430f59590c 100644
--- a/test/Bridges/Constraint/test_ScalarFunctionizeBridge.jl
+++ b/test/Bridges/Constraint/test_ScalarFunctionizeBridge.jl
@@ -246,17 +246,19 @@ function test_runtests()
         1.0 * x <= 1.0
         """,
     )
-    MOI.Bridges.runtests(
-        MOI.Bridges.Constraint.ScalarFunctionizeBridge,
-        """
-        variables: x
-        x in ZeroOne()
-        """,
-        """
-        variables: x
-        1.0 * x in ZeroOne()
-        """,
-    )
+    for set in ("ZeroOne()", "Integer()")
+        MOI.Bridges.runtests(
+            MOI.Bridges.Constraint.ScalarFunctionizeBridge,
+            """
+            variables: x
+            x in $set
+            """,
+            """
+            variables: x
+            1.0 * x in $set
+            """,
+        )
+    end
     MOI.Bridges.runtests(
         MOI.Bridges.Constraint.VectorFunctionizeBridge,
         """
@@ -290,6 +292,19 @@ function test_runtests()
         [1.0 * t, 1.0 * x] in SecondOrderCone(2)
         """,
     )
+    for set in ("SOS1([1.0, 2.0])", "SOS2([1.0, 2.0])")
+        MOI.Bridges.runtests(
+            MOI.Bridges.Constraint.VectorFunctionizeBridge,
+            """
+            variables: x, y
+            [x, y] in $set
+            """,
+            """
+            variables: x, y
+            [1.0 * x, 1.0 * y] in $set
+            """,
+        )
+    end
     return
 end
 
@@ -333,6 +348,27 @@ function test_FunctionConversionBridge()
     return
 end
 
+function test_FunctionConversionBridge_ConstraintDual()
+    inner = MOI.Utilities.MockOptimizer(
+        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
+    )
+    model = MOI.Bridges.Constraint.ToScalarNonlinear{Float64}(inner)
+    x, y = MOI.add_variables(model, 2)
+    f = 1.0 * x * x + 2.0 * x * y + 3.0 * y + 4.0
+    ci = MOI.add_constraint(model, f, MOI.GreaterThan(1.0))
+    bridge = MOI.get(model, MOI.Bridges.FirstBridge(), ci)
+    @test MOI.get(model, MOI.ConstraintFunction(), ci) ≈ f
+    @test MOI.get(inner, MOI.ConstraintFunction(), bridge.constraint) ≈
+          convert(MOI.ScalarNonlinearFunction, f)
+    @test MOI.supports(model, MOI.ConstraintDual(), typeof(ci))
+    for dual in (-1.2, 0.0, 1.2)
+        MOI.set(model, MOI.ConstraintDual(), ci, dual)
+        @test MOI.get(model, MOI.ConstraintDual(), ci) == dual
+        @test MOI.get(inner, MOI.ConstraintDual(), bridge.constraint) == dual
+    end
+    return
+end
+
 function test_FunctionConversionBridge_VectorNonlinearFunction()
     inner = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())
     BT = MOI.Bridges.Constraint.FunctionConversionBridge{
diff --git a/test/Bridges/Constraint/test_SplitHyperRectangleBridge.jl b/test/Bridges/Constraint/test_SplitHyperRectangleBridge.jl
index 33c39ccc98..f7f7a84e9e 100644
--- a/test/Bridges/Constraint/test_SplitHyperRectangleBridge.jl
+++ b/test/Bridges/Constraint/test_SplitHyperRectangleBridge.jl
@@ -95,6 +95,7 @@ function test_runtests_free_row()
         [1.0 * z + -3.0, 3.3 + -1.0 * z] in Nonnegatives(2)
         """;
         constraint_start = 3.0,
+        dual = [0.0, 3.0],
     )
     return
 end
@@ -108,7 +109,8 @@ function test_runtests_all_free_rows()
         """,
         """
         variables: x
-        """,
+        """;
+        dual = [0.0],
     )
     return
 end
diff --git a/test/Bridges/Variable/test_VectorizeBridge.jl b/test/Bridges/Variable/test_VectorizeBridge.jl
index 33936de912..3d44061b34 100644
--- a/test/Bridges/Variable/test_VectorizeBridge.jl
+++ b/test/Bridges/Variable/test_VectorizeBridge.jl
@@ -287,6 +287,39 @@ function test_runtests()
     return
 end
 
+function test_ConstraintDual_lagrangian_shift()
+    mock = MOI.Utilities.MockOptimizer(
+        MOI.Utilities.Model{Float64}();
+        eval_variable_constraint_dual = false,
+    )
+    model = MOI.Bridges.Variable.Vectorize{Float64}(mock)
+    x, _ = MOI.add_constrained_variable(model, MOI.GreaterThan(2.0))
+    MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)
+    MOI.set(
+        model,
+        MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
+        1.0 * x,
+    )
+    y = only(MOI.get(mock, MOI.ListOfVariableIndices()))
+    ci = only(
+        MOI.get(
+            mock,
+            MOI.ListOfConstraintIndices{
+                MOI.VectorOfVariables,
+                MOI.Nonnegatives,
+            }(),
+        ),
+    )
+    MOI.set(mock, MOI.VariablePrimal(), y, 3.0)
+    MOI.set(mock, MOI.ConstraintDual(), ci, [1.2])
+    MOI.set(mock, MOI.PrimalStatus(), MOI.FEASIBLE_POINT)
+    # The dual is deliberately nonstationary: the coefficient of y in the
+    # Lagrangian is 1 - 1.2. Both models should nevertheless give 5 - 3 * 1.2.
+    @test MOI.Bridges._test_lagrangian_value(model) ≈ 1.4
+    @test MOI.Bridges._test_lagrangian_value(mock) ≈ 1.4
+    return
+end
+
 function test_list_of_constraint_indices()
     inner = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())
     model = MOI.Bridges.Variable.Vectorize{Float64}(inner)
diff --git a/test/Utilities/test_mockoptimizer.jl b/test/Utilities/test_mockoptimizer.jl
index 1d297f72b6..d78f5f0045 100644
--- a/test/Utilities/test_mockoptimizer.jl
+++ b/test/Utilities/test_mockoptimizer.jl
@@ -191,6 +191,26 @@ function test_DualObjectiveValue()
     return
 end
 
+function test_ConstraintDual_discrete_sets()
+    mock = MOIU.MockOptimizer(
+        MOIU.UniversalFallback(MOIU.Model{Float64}());
+        eval_variable_constraint_dual = false,
+    )
+    x, y = MOI.add_variables(mock, 2)
+    for (f, set, dual) in (
+        (x, MOI.ZeroOne(), 1.2),
+        (y, MOI.Integer(), -2.3),
+        (MOI.VectorOfVariables([x, y]), MOI.SOS1([1.0, 2.0]), [1.2, -2.3]),
+        (MOI.VectorOfVariables([x, y]), MOI.SOS2([1.0, 2.0]), [-2.3, 1.2]),
+    )
+        ci = MOI.add_constraint(mock, f, set)
+        @test MOI.supports(mock, MOI.ConstraintDual(), typeof(ci))
+        MOI.set(mock, MOI.ConstraintDual(), ci, dual)
+        @test MOI.get(mock, MOI.ConstraintDual(), ci) == dual
+    end
+    return
+end
+
 function test_add_con_allowed_false()
     model = MOI.Utilities.MockOptimizer(
         MOI.Utilities.Model{Float64}();
diff --git a/test/Utilities/test_results.jl b/test/Utilities/test_results.jl
index 6b0c685dda..7e82bc5173 100644
--- a/test/Utilities/test_results.jl
+++ b/test/Utilities/test_results.jl
@@ -291,6 +291,62 @@ function test_get_fallback_DualObjectiveValue()
     return
 end
 
+function test_get_fallback_DualObjectiveValue_complex()
+    model = MOI.Utilities.MockOptimizer(
+        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
+    )
+    x = MOI.add_variable(model)
+    c1 = MOI.add_constraint(
+        model,
+        (1.0 + 2.0im) * x + (3.0 + 4.0im),
+        MOI.EqualTo(1.0 - 2.0im),
+    )
+    c2 = MOI.add_constraint(
+        model,
+        MOI.Utilities.vectorize([
+            (1.0 + 2.0im) * x + (3.0 + 4.0im),
+            -1.0im * x + (-2.0 + 5.0im),
+        ]),
+        MOI.Zeros(2),
+    )
+    MOI.set(model, MOI.ConstraintDual(), c1, 2.0 - 3.0im)
+    MOI.set(model, MOI.ConstraintDual(), c2, [2.0 - 3.0im, -1.0 + 4.0im])
+    MOI.set(model, MOI.DualStatus(), MOI.FEASIBLE_POINT)
+    for (sense, expected) in ((MOI.MIN_SENSE, -2.0), (MOI.MAX_SENSE, 2.0))
+        MOI.set(model, MOI.ObjectiveSense(), sense)
+        @test MOI.Utilities.get_fallback(
+            model,
+            MOI.DualObjectiveValue(),
+            Float64,
+        ) == expected
+    end
+    return
+end
+
+function test_get_fallback_DualObjectiveValue_Parameter()
+    model = MOI.Utilities.MockOptimizer(
+        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}());
+        eval_variable_constraint_dual = false,
+    )
+    x, ci = MOI.add_constrained_variable(model, MOI.Parameter(2.0))
+    MOI.set(
+        model,
+        MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
+        1.0 * x + 5.0,
+    )
+    MOI.set(model, MOI.ConstraintDual(), ci, 3.0)
+    MOI.set(model, MOI.DualStatus(), MOI.FEASIBLE_POINT)
+    for (sense, expected) in ((MOI.MIN_SENSE, 11.0), (MOI.MAX_SENSE, -1.0))
+        MOI.set(model, MOI.ObjectiveSense(), sense)
+        @test MOI.Utilities.get_fallback(
+            model,
+            MOI.DualObjectiveValue(),
+            Float64,
+        ) == expected
+    end
+    return
+end
+
 function test_get_fallback_DualObjectiveValue_Quadratic()
     T = Float64
     model = MOI.Utilities.MockOptimizer(
diff --git a/test/Utilities/test_set_dot.jl b/test/Utilities/test_set_dot.jl
index 969e0442c4..b9abe45406 100644
--- a/test/Utilities/test_set_dot.jl
+++ b/test/Utilities/test_set_dot.jl
@@ -53,6 +53,15 @@ function test_set_dot(T = Int)
     end
 end
 
+function test_set_dot_complex_triangle()
+    x = ComplexF64[1+2im, 3-4im, -2+5im]
+    y = ComplexF64[2-3im, -1+2im, 4+1im]
+    set = MOI.PositiveSemidefiniteConeTriangle(2)
+    @test MOI.Utilities.set_dot(x, y, set) == -29 - 25im
+    @test MOI.Utilities.set_dot(x, x, set) == 84
+    return
+end
+
 end
 
 TestSetDot.runtests()