diff --git a/src/MOI_wrapper.jl b/src/MOI_wrapper.jl
index d5ef9173..88bc3e82 100644
--- a/src/MOI_wrapper.jl
+++ b/src/MOI_wrapper.jl
@@ -519,11 +519,11 @@ function MOI.set(
     c::MOI.ConstraintIndex{MOI.ScalarQuadraticFunction{T},S},
     name::String,
 ) where {T,S<:MOI.AbstractSet}
+    c_aux = c
     if haskey(model.quadratic_outer_to_inner, c)
-        MOI.set(model.optimizer, attr, model.quadratic_outer_to_inner[c], name)
-    else
-        MOI.set(model.optimizer, attr, c, name)
+        c_aux = model.quadratic_outer_to_inner[c]
     end
+    MOI.set(model.optimizer, attr, c_aux, name)
     return
 end
 
@@ -533,16 +533,11 @@ function MOI.set(
     c::MOI.ConstraintIndex{MOI.VectorQuadraticFunction{T},S},
     name::String,
 ) where {T,S<:MOI.AbstractSet}
+    c_aux = c
     if haskey(model.vector_quadratic_outer_to_inner, c)
-        MOI.set(
-            model.optimizer,
-            attr,
-            model.vector_quadratic_outer_to_inner[c],
-            name,
-        )
-    else
-        MOI.set(model.optimizer, attr, c, name)
+        c_aux = model.vector_quadratic_outer_to_inner[c]
     end
+    MOI.set(model.optimizer, attr, c_aux, name)
     return
 end
 
@@ -565,11 +560,11 @@ function MOI.get(
     attr::MOI.ConstraintName,
     c::MOI.ConstraintIndex{MOI.ScalarQuadraticFunction{T},S},
 ) where {T,S<:MOI.AbstractSet}
+    c_aux = c
     if haskey(model.quadratic_outer_to_inner, c)
-        return MOI.get(model.optimizer, attr, model.quadratic_outer_to_inner[c])
-    else
-        return MOI.get(model.optimizer, attr, c)
+        c_aux = model.quadratic_outer_to_inner[c]
     end
+    return MOI.get(model.optimizer, attr, c_aux)
 end
 
 function MOI.get(
@@ -577,15 +572,11 @@ function MOI.get(
     attr::MOI.ConstraintName,
     c::MOI.ConstraintIndex{MOI.VectorQuadraticFunction{T},S},
 ) where {T,S<:MOI.AbstractSet}
+    c_aux = c
     if haskey(model.vector_quadratic_outer_to_inner, c)
-        return MOI.get(
-            model.optimizer,
-            attr,
-            model.vector_quadratic_outer_to_inner[c],
-        )
-    else
-        return MOI.get(model.optimizer, attr, c)
+        c_aux = model.vector_quadratic_outer_to_inner[c]
     end
+    return MOI.get(model.optimizer, attr, c_aux)
 end
 
 function MOI.get(
@@ -979,10 +970,6 @@ function _is_vector_affine(f::MOI.VectorQuadraticFunction{T}) where {T}
     return isempty(f.quadratic_terms)
 end
 
-function _is_vector_affine(::MOI.VectorAffineFunction{T}) where {T}
-    return true  # VectorAffineFunction is always affine
-end
-
 function _add_constraint_with_parameters_on_function(
     model::Optimizer,
     f::MOI.VectorQuadraticFunction{T},
@@ -1037,15 +1024,15 @@ function MOI.delete(
     model::Optimizer,
     c::MOI.ConstraintIndex{F,S},
 ) where {F<:MOI.VectorQuadraticFunction,S<:MOI.AbstractSet}
+    c_aux = c
     if haskey(model.vector_quadratic_outer_to_inner, c)
         ci_inner = model.vector_quadratic_outer_to_inner[c]
         delete!(model.vector_quadratic_outer_to_inner, c)
         delete!(model.vector_quadratic_constraint_cache, ci_inner)
         delete!(model.vector_quadratic_constraint_cache_set, ci_inner)
-        MOI.delete(model.optimizer, ci_inner)
-    else
-        MOI.delete(model.optimizer, c)
+        c_aux = ci_inner
     end
+    MOI.delete(model.optimizer, c_aux)
     delete!(model.constraint_outer_to_inner, c)
     return
 end
@@ -1054,15 +1041,15 @@ function MOI.delete(
     model::Optimizer,
     c::MOI.ConstraintIndex{F,S},
 ) where {F<:MOI.ScalarQuadraticFunction,S<:MOI.AbstractSet}
+    c_aux = c
     if haskey(model.quadratic_outer_to_inner, c)
         ci_inner = model.quadratic_outer_to_inner[c]
         delete!(model.quadratic_outer_to_inner, c)
         delete!(model.quadratic_constraint_cache, ci_inner)
         delete!(model.quadratic_constraint_cache_set, ci_inner)
-        MOI.delete(model.optimizer, ci_inner)
-    else
-        MOI.delete(model.optimizer, c)
+        c_aux = ci_inner
     end
+    MOI.delete(model.optimizer, c_aux)
     delete!(model.constraint_outer_to_inner, c)
     return
 end
@@ -1463,6 +1450,19 @@ function MOI.get(
     return model.parameters[p_idx(v)]
 end
 
+function MOI.get(
+    model::Optimizer,
+    ::MOI.ConstraintPrimalStart,
+    c::MOI.ConstraintIndex{MOI.VariableIndex,MOI.Parameter{T}},
+) where {T}
+    v = MOI.VariableIndex(c.value)
+    if _parameter_in_model(model, v)
+        return model.parameters[p_idx(v)]
+    else
+        error("Variable not in the model")
+    end
+end
+
 function MOI.set(
     model::Optimizer,
     ::MOI.ConstraintPrimalStart,
diff --git a/test/jump_tests.jl b/test/jump_tests.jl
index f3e3f6f3..d36e3ed6 100644
--- a/test/jump_tests.jl
+++ b/test/jump_tests.jl
@@ -1297,6 +1297,10 @@ function test_jump_psd_cone_with_parameter_pv()
         con,
         [0, (p * x - 1), 0] in MOI.PositiveSemidefiniteConeTriangle(2)
     )
+    @test MOI.get(backend(model), MOI.ConstraintFunction(), index(con)) isa
+          MOI.VectorQuadraticFunction{Float64}
+    @test MOI.get(backend(model), MOI.ConstraintSet(), index(con)) isa
+          MOI.PositiveSemidefiniteConeTriangle
     # the above constraint is equivalent to
     # - (p * x -1)^2 >=0 -> (p * x -1)^2 <= 0 -> (p * x -1) == 0 -> p*x == 1
     @test is_valid(model, con)
@@ -1439,13 +1443,13 @@ function test_jump_psd_cone_with_parameter_p_v_pv()
         con,
         [p, (2 * x - 3), p * 3 * x] in MOI.PositiveSemidefiniteConeTriangle(2)
     )
-    @objective(model, Min, x)
+    @objective(model, Min, x * x - p * p)
     @test is_valid(model, con)
     optimize!(model)
-    @test value(x) ≈ 0.7499854 atol = 1e-5
+    @test value(x) ≈ 0.7499854 atol = 1e-3
     set_parameter_value(p, 3.0)
     optimize!(model)
-    @test value(x) ≈ 0.236506 atol = 1e-5
+    @test value(x) ≈ 0.236506 atol = 1e-3
     return delete(model, con)
 end
 
@@ -1466,11 +1470,11 @@ function test_jump_psd_cone_with_parameter_p_v_pp()
         con,
         [p, (2 * x - 3), p * 3 * p] in MOI.PositiveSemidefiniteConeTriangle(2)
     )
-    @objective(model, Min, x)
+    @objective(model, Min, x - p)
     @test is_valid(model, con)
     optimize!(model)
     @test value(x) ≈ 0.633969 atol = 1e-5
-    set_parameter_value(p, 3.0)
+    set_parameter_value(p, Float32(3.0))
     optimize!(model)
     @test value(x) ≈ -2.9999734 atol = 1e-5
     return delete(model, con)
@@ -1561,6 +1565,17 @@ function test_variable_and_constraint_not_registered()
         MOI.ConstraintSet(),
         index(ParameterRef(p1)),
     )
+    @test_throws ErrorException("Variable not in the model") MOI.set(
+        backend(model2),
+        MOI.ConstraintPrimalStart(),
+        index(ParameterRef(p1)),
+        1.0,
+    )
+    @test_throws ErrorException("Variable not in the model") MOI.get(
+        backend(model2),
+        MOI.ConstraintPrimalStart(),
+        index(ParameterRef(p1)),
+    )
     @test_throws ErrorException("Variable not in the model") MOI.set(
         backend(model2),
         MOI.ObjectiveFunction{MOI.VariableIndex}(),
diff --git a/test/moi_tests.jl b/test/moi_tests.jl
index f5db78c6..9d81adfd 100644
--- a/test/moi_tests.jl
+++ b/test/moi_tests.jl
@@ -2073,3 +2073,14 @@ function test_psd_cone_with_parameter()
     MOI.set(model, MOI.ConstraintName(), c_index, "psd_cone")
     @test MOI.get(model, MOI.ConstraintName(), c_index) == "psd_cone"
 end
+
+function test_copy_model()
+    model = MOI.Utilities.Model{Float64}()
+    x = MOI.add_variable(model)
+    c = MOI.add_constraint(model, 1.0 * x, MOI.EqualTo(1.0))
+    MOI.set(model, MOI.ConstraintName(), c, "c")
+    poi = POI.Optimizer(GLPK.Optimizer())
+    MOI.copy_to(poi, model)
+    MOI.optimize!(poi)
+    @test MOI.get(poi, MOI.VariablePrimal(), x) ≈ 1.0
+end