diff --git a/src/Test/test_attribute.jl b/src/Test/test_attribute.jl
index 16948f37e8..93cae893cc 100644
--- a/src/Test/test_attribute.jl
+++ b/src/Test/test_attribute.jl
@@ -1,10 +1,10 @@
 """
-    test_attribute_NumberThreads(model::MOI.ModelLike, config::Config)
+    test_attribute_NumberThreads(model::MOI.AbstractOptimizer, config::Config)
 
 Test that the [`MOI.NumberOfThreads`](@ref) attribute is implemented for
 `model`.
 """
-function test_attribute_NumberThreads(model::MOI.ModelLike, ::Config)
+function test_attribute_NumberThreads(model::MOI.AbstractOptimizer, ::Config)
     @requires MOI.supports(model, MOI.NumberOfThreads())
     # Get the current value to restore it at the end of the test
     value = MOI.get(model, MOI.NumberOfThreads())
@@ -16,6 +16,7 @@ function test_attribute_NumberThreads(model::MOI.ModelLike, ::Config)
     @test value == MOI.get(model, MOI.NumberOfThreads())
     return
 end
+test_attribute_NumberThreads(::MOI.ModelLike, ::Config) = nothing
 
 function setup_test(
     ::typeof(test_attribute_NumberThreads),
@@ -27,12 +28,15 @@ function setup_test(
 end
 
 """
-    test_attribute_RawStatusString(model::MOI.ModelLike, config::Config)
+    test_attribute_RawStatusString(model::MOI.AbstractOptimizer, config::Config)
 
 Test that the [`MOI.RawStatusString`](@ref) attribute is implemented for
 `model`.
 """
-function test_attribute_RawStatusString(model::MOI.ModelLike, config::Config)
+function test_attribute_RawStatusString(
+    model::MOI.AbstractOptimizer,
+    config::Config,
+)
     @requires _supports(config, MOI.optimize!)
     @requires _supports(config, MOI.RawStatusString)
     MOI.add_variable(model)
@@ -40,6 +44,7 @@ function test_attribute_RawStatusString(model::MOI.ModelLike, config::Config)
     @test MOI.get(model, MOI.RawStatusString()) isa AbstractString
     return
 end
+test_attribute_RawStatusString(::MOI.ModelLike, ::Config) = nothing
 
 function setup_test(
     ::typeof(test_attribute_RawStatusString),
@@ -60,11 +65,11 @@ function setup_test(
 end
 
 """
-    test_attribute_Silent(model::MOI.ModelLike, config::Config)
+    test_attribute_Silent(model::MOI.AbstractOptimizer, config::Config)
 
 Test that the [`MOI.Silent`](@ref) attribute is implemented for `model`.
 """
-function test_attribute_Silent(model::MOI.ModelLike, ::Config)
+function test_attribute_Silent(model::MOI.AbstractOptimizer, ::Config)
     @requires MOI.supports(model, MOI.Silent())
     # Get the current value to restore it at the end of the test
     value = MOI.get(model, MOI.Silent())
@@ -77,6 +82,7 @@ function test_attribute_Silent(model::MOI.ModelLike, ::Config)
     @test value == MOI.get(model, MOI.Silent())
     return
 end
+test_attribute_Silent(::MOI.ModelLike, ::Config) = nothing
 
 function setup_test(
     ::typeof(test_attribute_Silent),
@@ -88,23 +94,27 @@ function setup_test(
 end
 
 """
-    test_attribute_SolverName(model::MOI.ModelLike, config::Config)
+    test_attribute_SolverName(model::MOI.AbstractOptimizer, config::Config)
 
 Test that the [`MOI.SolverName`](@ref) attribute is implemented for `model`.
 """
-function test_attribute_SolverName(model::MOI.ModelLike, config::Config)
+function test_attribute_SolverName(model::MOI.AbstractOptimizer, config::Config)
     if _supports(config, MOI.SolverName)
         @test MOI.get(model, MOI.SolverName()) isa AbstractString
     end
     return
 end
+test_attribute_SolverName(::MOI.ModelLike, ::Config) = nothing
 
 """
-    test_attribute_SolveTimeSec(model::MOI.ModelLike, config::Config)
+    test_attribute_SolveTimeSec(model::MOI.AbstractOptimizer, config::Config)
 
 Test that the [`MOI.SolveTimeSec`](@ref) attribute is implemented for `model`.
 """
-function test_attribute_SolveTimeSec(model::MOI.ModelLike, config::Config)
+function test_attribute_SolveTimeSec(
+    model::MOI.AbstractOptimizer,
+    config::Config,
+)
     @requires _supports(config, MOI.optimize!)
     @requires _supports(config, MOI.SolveTimeSec)
     MOI.add_variable(model)
@@ -112,6 +122,7 @@ function test_attribute_SolveTimeSec(model::MOI.ModelLike, config::Config)
     @test MOI.get(model, MOI.SolveTimeSec()) >= 0.0
     return
 end
+test_attribute_SolveTimeSec(::MOI.ModelLike, ::Config) = nothing
 
 function setup_test(
     ::typeof(test_attribute_SolveTimeSec),
@@ -126,11 +137,11 @@ function setup_test(
 end
 
 """
-    test_attribute_TimeLimitSec(model::MOI.ModelLike, config::Config)
+    test_attribute_TimeLimitSec(model::MOI.AbstractOptimizer, config::Config)
 
 Test that the [`MOI.TimeLimitSec`](@ref) attribute is implemented for `model`.
 """
-function test_attribute_TimeLimitSec(model::MOI.ModelLike, ::Config)
+function test_attribute_TimeLimitSec(model::MOI.AbstractOptimizer, ::Config)
     @requires MOI.supports(model, MOI.TimeLimitSec())
     # Get the current value to restore it at the end of the test
     value = MOI.get(model, MOI.TimeLimitSec())
@@ -142,6 +153,7 @@ function test_attribute_TimeLimitSec(model::MOI.ModelLike, ::Config)
     @test value == MOI.get(model, MOI.TimeLimitSec()) # Equality should hold
     return
 end
+test_attribute_TimeLimitSec(::MOI.ModelLike, ::Config) = nothing
 
 function setup_test(
     ::typeof(test_attribute_TimeLimitSec),
diff --git a/src/Test/test_constraint.jl b/src/Test/test_constraint.jl
index d45c928bb7..4ab7b8802f 100644
--- a/src/Test/test_constraint.jl
+++ b/src/Test/test_constraint.jl
@@ -728,6 +728,11 @@ function test_constraint_ConstraintPrimalStart(
         MOI.VectorAffineFunction{T},
         MOI.Nonnegatives,
     )
+    @requires MOI.supports(
+        model,
+        MOI.ConstraintPrimalStart(),
+        MOI.ConstraintIndex{MOI.VectorAffineFunction{T},MOI.Nonnegatives},
+    )
     x = MOI.add_variable(model)
     f = MOI.VectorAffineFunction(
         [MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(one(T), x))],
@@ -759,6 +764,11 @@ function test_constraint_ConstraintDualStart(
         MOI.VectorAffineFunction{T},
         MOI.Nonnegatives,
     )
+    @requires MOI.supports(
+        model,
+        MOI.ConstraintDualStart(),
+        MOI.ConstraintIndex{MOI.VectorAffineFunction{T},MOI.Nonnegatives},
+    )
     x = MOI.add_variable(model)
     f = MOI.VectorAffineFunction(
         [MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(one(T), x))],
diff --git a/src/Test/test_model.jl b/src/Test/test_model.jl
index 3188915079..62d8cea500 100644
--- a/src/Test/test_model.jl
+++ b/src/Test/test_model.jl
@@ -742,11 +742,10 @@ end
         config::Config,
     )
 
-Test that `MOI.ScalarFunctionConstantNotZero` is thrown when a constraint with
-a function with nonzero constant is added.
+Test adding a linear constraint with a non-zero function constant.
 
-**This test is optional because solvers could choose to support scalar functions
-with nonzero constants.**
+This should either work, or error with `MOI.ScalarFunctionConstantNotZero` if
+the model does not support it.
 
 To skip this test, pass `MOI.ScalarFunctionConstantNotZero` to the `exclude`
 argument of [`Config`](@ref).
@@ -756,33 +755,30 @@ function test_model_ScalarFunctionConstantNotZero(
     config::Config,
 )
     @requires _supports(config, MOI.ScalarFunctionConstantNotZero)
-    err = MOI.ScalarFunctionConstantNotZero{
-        Float64,
-        MOI.ScalarAffineFunction{Float64},
-        MOI.EqualTo{Float64},
-    }(
-        1.0,
-    )
-    @test_throws err begin
-        MOI.add_constraint(
-            model,
-            MOI.ScalarAffineFunction(MOI.ScalarAffineTerm{Float64}[], 1.0),
-            MOI.EqualTo(2.0),
+    function _error(S, value)
+        return MOI.ScalarFunctionConstantNotZero{
+            Float64,
+            MOI.ScalarAffineFunction{Float64},
+            S,
+        }(
+            value,
         )
     end
-    err = MOI.ScalarFunctionConstantNotZero{
-        Float64,
-        MOI.ScalarAffineFunction{Float64},
-        MOI.GreaterThan{Float64},
-    }(
-        2.0,
-    )
-    @test_throws err begin
-        MOI.add_constraint(
-            model,
-            MOI.ScalarAffineFunction(MOI.ScalarAffineTerm{Float64}[], 2.0),
-            MOI.GreaterThan(1.0),
-        )
+    try
+        f = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm{Float64}[], 1.0)
+        c = MOI.add_constraint(model, f, MOI.EqualTo(2.0))
+        @requires _supports(config, MOI.ConstraintFunction)
+        @test MOI.get(model, MOI.ConstraintFunction(), c) ≈ f
+    catch err
+        @test err == _error(MOI.EqualTo{Float64}, 1.0)
+    end
+    try
+        f = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm{Float64}[], 2.0)
+        c = MOI.add_constraint(model, f, MOI.GreaterThan(1.0))
+        @requires _supports(config, MOI.ConstraintFunction)
+        @test MOI.get(model, MOI.ConstraintFunction(), c) ≈ f
+    catch err
+        @test err == _error(MOI.GreaterThan{Float64}, 2.0)
     end
     return
 end
diff --git a/test/Utilities/cachingoptimizer.jl b/test/Utilities/cachingoptimizer.jl
index 5c530c9b77..4b70ac4905 100644
--- a/test/Utilities/cachingoptimizer.jl
+++ b/test/Utilities/cachingoptimizer.jl
@@ -114,14 +114,7 @@ function test_MOI_Test()
             MOI.Test.runtests(
                 model,
                 MOI.Test.Config(exclude = Any[MOI.optimize!]),
-                exclude = String[
-                    # This test is optional.
-                    "test_model_ScalarFunctionConstantNotZero",
-                    # MockOptimizer doesnt' support these
-                    "test_attribute_SolverName",
-                    "test_constraint_ConstraintDualStart",
-                    "test_constraint_ConstraintPrimalStart",
-                ],
+                exclude = ["test_attribute_SolverName"],
             )
         end
     end
diff --git a/test/Utilities/model.jl b/test/Utilities/model.jl
index eeb7532c0f..184b257eac 100644
--- a/test/Utilities/model.jl
+++ b/test/Utilities/model.jl
@@ -87,15 +87,7 @@ struct SetNotSupportedBySolvers <: MOI.AbstractSet end
 function test_MOI_Test()
     MOI.Test.runtests(
         MOI.Utilities.Model{Float64}(),
-        MOI.Test.Config(exclude = Any[MOI.optimize!]),
-        exclude = String[
-            # This test is optional.
-            "test_model_ScalarFunctionConstantNotZero",
-            # Unsupported attributes
-            "test_constraint_ConstraintPrimalStart",
-            "test_constraint_ConstraintDualStart",
-            "test_attribute_SolverName",
-        ],
+        MOI.Test.Config(exclude = Any[MOI.optimize!,]),
     )
     return
 end
diff --git a/test/Utilities/universalfallback.jl b/test/Utilities/universalfallback.jl
index cde177e20d..704660f68f 100644
--- a/test/Utilities/universalfallback.jl
+++ b/test/Utilities/universalfallback.jl
@@ -71,8 +71,6 @@ function test_MOI_Test()
         model,
         MOI.Test.Config(exclude = Any[MOI.optimize!]),
         exclude = String[
-            # This test is optional.
-            "test_model_ScalarFunctionConstantNotZero",
             # UniversalFallback fails all these tests because it supports
             # everything...
             "test_attribute_",