diff --git a/src/MOI_wrapper.jl b/src/MOI_wrapper.jl
index 71ac674..c1f5f0c 100644
--- a/src/MOI_wrapper.jl
+++ b/src/MOI_wrapper.jl
@@ -1950,6 +1950,7 @@ end
 
 function MOI.optimize!(model::Optimizer)
     if !isempty(model.updated_parameters)
+        MOI.Utilities.final_touch(model, nothing)
         update_parameters!(model)
     end
     MOI.optimize!(model.optimizer)
diff --git a/src/ParametricOptInterface.jl b/src/ParametricOptInterface.jl
index ce0d928..5ade1c4 100644
--- a/src/ParametricOptInterface.jl
+++ b/src/ParametricOptInterface.jl
@@ -57,13 +57,17 @@ include("parametric_cubic_function.jl")
         evaluate_duals::Bool = true,
         save_original_objective_and_constraints::Bool = true,
         with_bridge_type = nothing,
+        with_cache_type = nothing,
     )
 
 Create an `Optimizer`, which allows the handling of parameters in an
 optimization model.
 
 If `optimizer` is not a `MOI.ModelLike,` the inner optimizer is constructed
-using `MOI.instantiate(optimizer; with_bridge_type)`.
+using `MOI.instantiate(optimizer; with_cache_type)`.
+
+If `with_bridge_type !== nothing`, a `MOI.Bridges.full_bridge_optimizer` is
+applied as an outer layer.
 
 The `{T}` type parameter is optional; it defaults to `Float64`.
 
@@ -82,8 +86,9 @@ The `{T}` type parameter is optional; it defaults to `Float64`.
   Note that this might break printing or queries such as
   `MOI.get(model, MOI.ConstraintFunction(), c)`. Defaults to `true`.
 
-- `with_bridge_type`: this is ignroed if `optimizer::MOI.ModelLike`, otherwise
-  it is passed to `MOI.instantiate`.
+- `with_bridge_type`: the type passed to `MOI.Bridges.full_bridge_optimizer`
+
+- `with_cache_type`: the type passed to `MOI.instantiate`
 
 ## Example
 
@@ -276,13 +281,24 @@ Optimizer(arg; kwargs...) = Optimizer{Float64}(arg; kwargs...)
 function Optimizer{T}(
     optimizer_fn;
     with_bridge_type = nothing,
+    with_cache_type = nothing,
     kwargs...,
 ) where {T}
-    inner = MOI.instantiate(optimizer_fn; with_bridge_type)
+    inner = MOI.instantiate(optimizer_fn; with_cache_type)
     if !MOI.supports_incremental_interface(inner)
-        cache = MOI.default_cache(inner, T)
+        # Don't use `default_cache` for the cache because, for example, SCS's
+        # default cache doesn't support modifying coefficients of the constraint
+        # matrix. JuMP uses the default cache with SCS because it has an outer
+        # layer of caching; we don't have that here, so we can't use the
+        # default.
+        #
+        # We could revert to using the default cache if we fix this in MOI.
+        cache = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{T}())
         inner = MOI.Utilities.CachingOptimizer(cache, inner)
     end
+    if with_bridge_type !== nothing
+        inner = MOI.Bridges.full_bridge_optimizer(inner, with_bridge_type)
+    end
     return Optimizer{T}(inner; kwargs...)
 end
 
diff --git a/test/Project.toml b/test/Project.toml
index 4adebdb..3642db5 100644
--- a/test/Project.toml
+++ b/test/Project.toml
@@ -12,4 +12,4 @@ Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 HiGHS = "1"
 Ipopt = "1"
 JuMP = "1"
-SCS = "1"
+SCS = "2"
diff --git a/test/test_JuMP.jl b/test/test_JuMP.jl
index df6d176..7a1e6c2 100644
--- a/test/test_JuMP.jl
+++ b/test/test_JuMP.jl
@@ -834,12 +834,7 @@ function test_jump_dual_delete_constraint_2()
 end
 
 function test_jump_dual_delete_constraint_3()
-    cached = MOI.Utilities.CachingOptimizer(
-        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-        SCS.Optimizer(),
-    )
-    optimizer = POI.Optimizer(cached)
-    model = direct_model(optimizer)
+    model = direct_model(POI.Optimizer(SCS.Optimizer))
     set_silent(model)
     list = []
     @variable(model, α in Parameter(1.0))
@@ -1211,15 +1206,8 @@ function test_parameter_Cannot_be_inf_2()
 end
 
 function test_jump_psd_cone_with_parameter_pv()
-    cached = MOI.Bridges.full_bridge_optimizer(
-        MOI.Utilities.CachingOptimizer(
-            MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-            SCS.Optimizer(),
-        ),
-        Float64,
-    )
-    optimizer = POI.Optimizer(cached)
-    model = direct_model(optimizer)
+    inner = POI.Optimizer(SCS.Optimizer; with_bridge_type = Float64)
+    model = direct_model(inner)
     set_silent(model)
     @variable(model, x)
     @variable(model, p in Parameter(1.0))
@@ -1406,18 +1394,8 @@ function test_jump_psd_cone_without_parameter_v_and_vv()
 end
 
 function test_variable_and_constraint_not_registered()
-    cached1 = MOI.Utilities.CachingOptimizer(
-        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-        SCS.Optimizer(),
-    )
-    optimizer1 = POI.Optimizer(cached1)
-    model1 = direct_model(optimizer1)
-    cached2 = MOI.Utilities.CachingOptimizer(
-        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-        SCS.Optimizer(),
-    )
-    optimizer2 = POI.Optimizer(cached2)
-    model2 = direct_model(optimizer2)
+    model1 = direct_model(POI.Optimizer(SCS.Optimizer))
+    model2 = direct_model(POI.Optimizer(SCS.Optimizer))
     set_silent(model1)
     set_silent(model2)
     @variable(model1, x)
@@ -1505,12 +1483,7 @@ function test_variable_and_constraint_not_registered()
 end
 
 function test_jump_errors()
-    cached1 = MOI.Utilities.CachingOptimizer(
-        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-        SCS.Optimizer(),
-    )
-    optimizer1 = POI.Optimizer(cached1)
-    model = direct_model(optimizer1)
+    model = direct_model(POI.Optimizer(SCS.Optimizer))
     @test_throws MOI.UnsupportedAttribute MOI.get(
         backend(model),
         MOI.NLPBlock(),
diff --git a/test/test_MathOptInterface.jl b/test/test_MathOptInterface.jl
index 8fc7167..82fc5ea 100644
--- a/test/test_MathOptInterface.jl
+++ b/test/test_MathOptInterface.jl
@@ -322,12 +322,10 @@ function test_moi_highs()
 end
 
 function test_moi_ipopt()
-    model = MOI.Utilities.CachingOptimizer(
-        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-        MOI.Bridges.full_bridge_optimizer(
-            POI.Optimizer(Ipopt.Optimizer),
-            Float64,
-        ),
+    model = POI.Optimizer(
+        Ipopt.Optimizer;
+        with_bridge_type = Float64,
+        with_cache_type = Float64,
     )
     MOI.set(model, MOI.Silent(), true)
     # Without fixed_variable_treatment set, duals are not computed for variables
@@ -364,6 +362,7 @@ function test_moi_ipopt()
             #  - CachingOptimizer does not throw if optimizer not attached
             "test_model_copy_to_UnsupportedAttribute",
             "test_model_copy_to_UnsupportedConstraint",
+            "test_model_ModelFilter_AbstractConstraintAttribute",
             #  - POI only supports cubic polynomial ScalarNonlinearFunction
             "test_nonlinear_duals",
             "test_nonlinear_expression_",
@@ -570,14 +569,7 @@ function test_production_problem_example_duals()
 end
 
 function test_production_problem_example_parameters_for_duals_and_intervals()
-    cached = MOI.Bridges.full_bridge_optimizer(
-        MOI.Utilities.CachingOptimizer(
-            MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-            HiGHS.Optimizer(),
-        ),
-        Float64,
-    )
-    optimizer = POI.Optimizer(cached)
+    optimizer = POI.Optimizer(HiGHS.Optimizer; with_bridge_type = Float64)
     MOI.set(optimizer, MOI.Silent(), true)
     c = [4.0, 3.0]
     A1 = [2.0, 1.0, 3.0]
@@ -639,14 +631,14 @@ function test_production_problem_example_parameters_for_duals_and_intervals()
     MOI.set(optimizer, MOI.ConstraintSet(), cz, MOI.Parameter(1.0))
     MOI.optimize!(optimizer)
     @test ≈(MOI.get(optimizer, MOI.ObjectiveValue()), 7.0, atol = ATOL)
-    @test MOI.get.(optimizer, MOI.VariablePrimal(), x) == [0.0, 1.0]
+    @test MOI.get.(optimizer, MOI.VariablePrimal(), x) ≈ [0.0, 1.0]
     @test ≈(MOI.get(optimizer, MOI.ConstraintDual(), cy), 9.0, atol = ATOL)
     @test ≈(MOI.get(optimizer, MOI.ConstraintDual(), cz), 0.0, atol = ATOL)
     @test ≈(MOI.get(optimizer, MOI.ConstraintDual(), cw), -2.0, atol = ATOL)
     MOI.set(optimizer, MOI.ConstraintSet(), cw, MOI.Parameter(0.0))
     MOI.optimize!(optimizer)
     @test ≈(MOI.get(optimizer, MOI.ObjectiveValue()), 3.0, atol = ATOL)
-    @test MOI.get.(optimizer, MOI.VariablePrimal(), x) == [0.0, 1.0]
+    @test MOI.get.(optimizer, MOI.VariablePrimal(), x) ≈ [0.0, 1.0]
     @test ≈(MOI.get(optimizer, MOI.ConstraintDual(), cy), 9.0, atol = ATOL)
     @test ≈(MOI.get(optimizer, MOI.ConstraintDual(), cz), 0.0, atol = ATOL)
     @test ≈(MOI.get(optimizer, MOI.ConstraintDual(), cw), -2.0, atol = ATOL)
@@ -2035,6 +2027,7 @@ function test_psd_cone_with_parameter()
     @test MOI.get(model, MOI.ConstraintName(), c_index) == ""
     MOI.set(model, MOI.ConstraintName(), c_index, "psd_cone")
     @test MOI.get(model, MOI.ConstraintName(), c_index) == "psd_cone"
+    return
 end
 
 function test_copy_model()
@@ -2125,15 +2118,13 @@ end
 struct VariableAttributeForTest <: MOI.AbstractVariableAttribute end
 struct ConstraintAttributeForTest <: MOI.AbstractConstraintAttribute end
 
-function test_variable_attribute_error()
-    solver = HiGHS.Optimizer()
-    MOI.set(solver, MOI.Silent(), true)
+function test_AA_variable_attribute_error()
     model = POI.Optimizer(
-        MOI.Utilities.CachingOptimizer(
-            MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-            MOI.Bridges.full_bridge_optimizer(solver, Float64),
-        ),
+        HiGHS.Optimizer;
+        with_bridge_type = Float64,
+        with_cache_type = Float64,
     )
+    MOI.set(model, MOI.Silent(), true)
     x = MOI.add_variable(model)
     MOI.set(model, VariableAttributeForTest(), x, 1.0)
     p, pc = MOI.add_constrained_variable(model, MOI.Parameter(1.0))
@@ -2148,13 +2139,8 @@ function test_variable_attribute_error()
 end
 
 function test_constraint_attribute_error()
-    solver = MOI.Utilities.Model{Float64}()
-    model = POI.Optimizer(
-        MOI.Utilities.CachingOptimizer(
-            MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-            MOI.Bridges.full_bridge_optimizer(solver, Float64),
-        ),
-    )
+    model =
+        POI.Optimizer(MOI.Utilities.Model{Float64}; with_bridge_type = Float64)
     MOI.supports(
         model,
         ConstraintAttributeForTest(),
@@ -2170,13 +2156,8 @@ function test_constraint_attribute_error()
 end
 
 function test_name_from_bound()
-    solver = MOI.Utilities.Model{Float64}()
-    model = POI.Optimizer(
-        MOI.Utilities.CachingOptimizer(
-            MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-            MOI.Bridges.full_bridge_optimizer(solver, Float64),
-        ),
-    )
+    model =
+        POI.Optimizer(MOI.Utilities.Model{Float64}; with_bridge_type = Float64)
     x = MOI.add_variable(model)
     p, pc = MOI.add_constrained_variable(model, MOI.Parameter(1.0))
     MOI.set(model, POI.ConstraintsInterpretation(), POI.ONLY_BOUNDS)
@@ -2200,15 +2181,12 @@ function test_get_constraint_set()
 end
 
 function test_quadratic_variable_parameter()
-    # model = POI.Optimizer(MOI.Utilities.Model{Float64}())
-    solver = Ipopt.Optimizer()
-    MOI.set(solver, MOI.Silent(), true)
     model = POI.Optimizer(
-        MOI.Utilities.CachingOptimizer(
-            MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
-            MOI.Bridges.full_bridge_optimizer(solver, Float64),
-        ),
+        Ipopt.Optimizer;
+        with_bridge_type = Float64,
+        with_cache_type = Float64,
     )
+    MOI.set(model, MOI.Silent(), true)
     x = MOI.add_variable(model)
     p, pc = MOI.add_constrained_variable(model, MOI.Parameter(1.0))
     f = 1.0 * x * x - 2.0 * p * p