diff --git a/src/MOI_wrapper.jl b/src/MOI_wrapper.jl
index 6d9e520c..70f98805 100644
--- a/src/MOI_wrapper.jl
+++ b/src/MOI_wrapper.jl
@@ -247,12 +247,13 @@ function MOI.get(model::Optimizer, tp::Type{MOI.VariableIndex}, attr::String)
     return MOI.get(model.optimizer, tp, attr)
 end
 
-function MOI.add_variable(model::Optimizer)
+function _add_variable(model::Optimizer, inner_vi)
     _next_variable_index!(model)
-    return MOI.Utilities.CleverDicts.add_item(
-        model.variables,
-        MOI.add_variable(model.optimizer),
-    )
+    return MOI.Utilities.CleverDicts.add_item(model.variables, inner_vi)
+end
+
+function MOI.add_variable(model::Optimizer)
+    return _add_variable(model, MOI.add_variable(model.optimizer))
 end
 
 function MOI.supports_add_constrained_variable(
@@ -269,6 +270,20 @@ function MOI.supports_add_constrained_variables(
     return MOI.supports_add_constrained_variables(model.optimizer, MOI.Reals)
 end
 
+function MOI.supports_add_constrained_variable(
+    model::Optimizer,
+    ::Type{S},
+) where {S<:MOI.AbstractScalarSet}
+    return MOI.supports_add_constrained_variable(model.optimizer, S)
+end
+
+function MOI.supports_add_constrained_variables(
+    model::Optimizer,
+    ::Type{S},
+) where {S<:MOI.AbstractVectorSet}
+    return MOI.supports_add_constrained_variables(model.optimizer, S)
+end
+
 function _assert_parameter_is_finite(set::MOI.Parameter{T}) where {T}
     if !isfinite(set.value)
         throw(
@@ -296,6 +311,27 @@ function MOI.add_constrained_variable(
     return p, cp
 end
 
+function MOI.add_constrained_variable(
+    model::Optimizer,
+    set::MOI.AbstractScalarSet,
+)
+    inner_vi, inner_ci = MOI.add_constrained_variable(model.optimizer, set)
+    outer_vi = _add_variable(model, inner_vi)
+    outer_ci =
+        MOI.ConstraintIndex{MOI.VariableIndex,typeof(set)}(outer_vi.value)
+    model.constraint_outer_to_inner[outer_ci] = inner_ci
+    return outer_vi, outer_ci
+end
+
+function MOI.add_constrained_variables(
+    model::Optimizer,
+    set::MOI.AbstractVectorSet,
+)
+    inner_vis, inner_ci = MOI.add_constrained_variables(model.optimizer, set)
+    _add_to_constraint_map!(model, inner_ci)
+    return _add_variable.(model, inner_vis), inner_ci
+end
+
 function _add_to_constraint_map!(model::Optimizer, ci)
     model.constraint_outer_to_inner[ci] = ci
     return
diff --git a/src/ParametricOptInterface.jl b/src/ParametricOptInterface.jl
index c1ba6942..32710300 100644
--- a/src/ParametricOptInterface.jl
+++ b/src/ParametricOptInterface.jl
@@ -180,12 +180,11 @@ mutable struct Optimizer{T,OT<:MOI.ModelLike} <: MOI.AbstractOptimizer
 
     # extension data
     ext::Dict{Symbol,Any}
-    function Optimizer(
+    function Optimizer{T}(
         optimizer::OT;
         evaluate_duals::Bool = true,
         save_original_objective_and_constraints::Bool = true,
-    ) where {OT}
-        T = Float64
+    ) where {T,OT}
         return new{T,OT}(
             optimizer,
             MOI.Utilities.CleverDicts.CleverDict{ParameterIndex,T}(
@@ -248,6 +247,8 @@ mutable struct Optimizer{T,OT<:MOI.ModelLike} <: MOI.AbstractOptimizer
     end
 end
 
+Optimizer(args...; kws...) = Optimizer{Float64}(args...; kws...)
+
 function _next_variable_index!(model::Optimizer)
     return model.last_variable_index_added += 1
 end
diff --git a/test/moi_tests.jl b/test/moi_tests.jl
index 524f9914..0993bbe2 100644
--- a/test/moi_tests.jl
+++ b/test/moi_tests.jl
@@ -2097,7 +2097,7 @@ function test_psd_cone_with_parameter()
     model = POI.Optimizer(cached)
     MOI.set(model, MOI.Silent(), true)
     x = MOI.add_variable(model)
-    p = first.(MOI.add_constrained_variable.(model, MOI.Parameter(1.0)),)
+    p = first.(MOI.add_constrained_variable.(model, MOI.Parameter(1.0)))
 
     # Set objective: minimize x
     obj_func = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0)
@@ -2142,3 +2142,63 @@ function test_copy_model()
     MOI.optimize!(poi)
     @test MOI.get(poi, MOI.VariablePrimal(), x) ≈ 1.0
 end
+
+function test_constrained_variable_glpk()
+    optimizer = POI.Optimizer(GLPK.Optimizer())
+    MOI.set(optimizer, MOI.Silent(), true)
+    set = MOI.LessThan(1.0)
+    @test MOI.supports_add_constrained_variable(optimizer, typeof(set))
+    x, c = MOI.add_constrained_variable(optimizer, set)
+    @test x.value == c.value
+    @test c isa MOI.ConstraintIndex{typeof(x),typeof(set)}
+    @test MOI.get(optimizer, MOI.ConstraintFunction(), c) ≈ x
+    @test MOI.get(optimizer, MOI.ConstraintSet(), c) == set
+    @test MOI.supports(optimizer, MOI.VariableName(), typeof(x))
+    MOI.set(optimizer, MOI.VariableName(), x, "vname")
+    @test MOI.get(optimizer, MOI.VariableName(), x) == "vname"
+    MOI.set(optimizer, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+    obj_func = 1.0 * x
+    MOI.set(optimizer, MOI.ObjectiveFunction{typeof(obj_func)}(), obj_func)
+    MOI.optimize!(optimizer)
+    @test MOI.get(optimizer, MOI.ConstraintDual(), c) ≈ -1 atol = ATOL
+    @test MOI.get(optimizer, MOI.VariablePrimal(), x) ≈ 1 atol = ATOL
+    return
+end
+
+include("no_free_model.jl")
+
+function test_constrained_variable_no_free()
+    optimizer = POI.Optimizer(NoFreeVariablesModel{Float64}())
+    set = MOI.LessThan(1.0)
+    @test MOI.supports_add_constrained_variable(optimizer, typeof(set))
+    x, c = MOI.add_constrained_variable(optimizer, set)
+    @test c isa MOI.ConstraintIndex{typeof(x),typeof(set)}
+    @test x.value == c.value
+    @test MOI.get(optimizer, MOI.ConstraintFunction(), c) ≈ x
+    @test MOI.get(optimizer, MOI.ConstraintSet(), c) == set
+    @test MOI.supports(optimizer, MOI.VariableName(), typeof(x))
+    MOI.set(optimizer, MOI.VariableName(), x, "vname")
+    @test MOI.get(optimizer, MOI.VariableName(), x) == "vname"
+    MOI.set(optimizer, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+    obj_func = 1.0 * x
+    MOI.set(optimizer, MOI.ObjectiveFunction{typeof(obj_func)}(), obj_func)
+    return
+end
+
+function test_constrained_variables()
+    optimizer = POI.Optimizer{Int}(NoFreeVariablesModel{Int}())
+    set = MOI.Nonnegatives(2)
+    @test MOI.supports_add_constrained_variables(optimizer, typeof(set))
+    x, c = MOI.add_constrained_variables(optimizer, set)
+    @test c isa MOI.ConstraintIndex{MOI.VectorOfVariables,typeof(set)}
+    @test MOI.get(optimizer, MOI.ConstraintFunction(), c) ≈
+          MOI.VectorOfVariables(x)
+    @test MOI.get(optimizer, MOI.ConstraintSet(), c) == set
+    @test MOI.supports(optimizer, MOI.VariableName(), eltype(x))
+    MOI.set(optimizer, MOI.VariableName(), x[1], "vname")
+    @test MOI.get(optimizer, MOI.VariableName(), x[1]) == "vname"
+    MOI.set(optimizer, MOI.ObjectiveSense(), MOI.MIN_SENSE)
+    obj_func = 1 * x[1] + 1 * x[2]
+    MOI.set(optimizer, MOI.ObjectiveFunction{typeof(obj_func)}(), obj_func)
+    return
+end
diff --git a/test/no_free_model.jl b/test/no_free_model.jl
new file mode 100644
index 00000000..c172a222
--- /dev/null
+++ b/test/no_free_model.jl
@@ -0,0 +1,40 @@
+MOI.Utilities.@model(
+    NoFreeVariablesModel,
+    (),
+    (),
+    (MOI.Nonnegatives,),
+    (),
+    (),
+    (),
+    (MOI.VectorOfVariables,),
+    (),
+)
+
+function MOI.supports_constraint(
+    ::NoFreeVariablesModel,
+    ::Type{MOI.VectorOfVariables},
+    ::Type{MOI.Reals},
+)
+    return false
+end
+
+function MOI.supports_add_constrained_variable(
+    ::NoFreeVariablesModel{T},
+    ::Type{MOI.LessThan{T}},
+) where {T}
+    return true
+end
+
+function MOI.supports_add_constrained_variables(
+    ::NoFreeVariablesModel,
+    ::Type{MOI.Nonnegatives},
+)
+    return true
+end
+
+function MOI.supports_add_constrained_variables(
+    ::NoFreeVariablesModel,
+    ::Type{MOI.Reals},
+)
+    return false
+end