diff --git a/src/MOI_wrapper/objective.jl b/src/MOI_wrapper/objective.jl
index 31c56fc1..51c059ec 100644
--- a/src/MOI_wrapper/objective.jl
+++ b/src/MOI_wrapper/objective.jl
@@ -1,7 +1,12 @@
-# objective function & sense
+# Objective function & sense.
+#
+# SCIP only supports affine objectives. For quadratic or nonlinear objectives,
+# the solver will depend on bridges with auxiliary variables. Single variable
+# objectives are also accepted, but the type is not correctly remembered.
 
 MOI.supports(::Optimizer, ::MOI.ObjectiveSense) = true
 MOI.supports(::Optimizer, ::MOI.ObjectiveFunction{SAF}) = true
+MOI.supports(::Optimizer, ::MOI.ObjectiveFunction{SVF}) = true
 
 function MOI.set(o::Optimizer, ::MOI.ObjectiveFunction{SAF}, obj::SAF)
     allow_modification(o)
@@ -24,6 +29,11 @@ function MOI.set(o::Optimizer, ::MOI.ObjectiveFunction{SAF}, obj::SAF)
     return nothing
 end
 
+function MOI.set(o::Optimizer, ::MOI.ObjectiveFunction{SVF}, obj::SVF)
+    aff_obj = SAF([AFF_TERM(1.0, obj.variable)], 0.0)
+    return MOI.set(o, MOI.ObjectiveFunction{SAF}(), aff_obj)
+end
+
 function MOI.get(o::Optimizer, ::MOI.ObjectiveFunction{SAF})
     terms = AFF_TERM[]
     for vr = keys(o.mscip.vars)
@@ -35,6 +45,16 @@ function MOI.get(o::Optimizer, ::MOI.ObjectiveFunction{SAF})
     return SAF(terms, constant)
 end
 
+function MOI.get(o::Optimizer, ::MOI.ObjectiveFunction{SVF})
+    aff_obj = MOI.get(o, MOI.ObjectiveFunction{SAF}())
+    if (length(aff_obj.terms) != 1
+        || aff_obj.terms[1].coefficient != 1.0
+        || aff_obj.constant != 0.0)
+        error("Objective is not single variable: $aff_obj !")
+    end
+    return SVF(aff_obj.terms[1].variable_index)
+end
+
 function MOI.set(o::Optimizer, ::MOI.ObjectiveSense, sense::MOI.OptimizationSense)
     allow_modification(o)
     if sense == MOI.MIN_SENSE
diff --git a/test/MINLPTests/run_minlptests.jl b/test/MINLPTests/run_minlptests.jl
index e9d9dd1c..a6bdebf0 100644
--- a/test/MINLPTests/run_minlptests.jl
+++ b/test/MINLPTests/run_minlptests.jl
@@ -19,6 +19,11 @@ const DUALTOL = NaN  # to disable the query
 end
 
 @testset "MINLPTests - nlp cvx" begin
+    @testset "nlp_cvx_001_010" begin
+        MINLPTests.nlp_cvx_001_010(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
     @testset "nlp_cvx_002_010" begin
         MINLPTests.nlp_cvx_002_010(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
                                    termination_target = MOI.OPTIMAL)
@@ -29,6 +34,16 @@ end
                                    termination_target = MOI.OPTIMAL)
     end
 
+    @testset "nlp_cvx_101_012" begin
+        MINLPTests.nlp_cvx_101_012(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
+    @testset "nlp_cvx_102_010" begin
+        MINLPTests.nlp_cvx_102_010(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
     @testset "nlp_cvx_102_011" begin
         MINLPTests.nlp_cvx_102_011(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
                                    termination_target = MOI.OPTIMAL)
@@ -39,6 +54,16 @@ end
                                    termination_target = MOI.OPTIMAL)
     end
 
+    @testset "nlp_cvx_103_010" begin
+        MINLPTests.nlp_cvx_103_010(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
+    @testset "nlp_cvx_103_011" begin
+        MINLPTests.nlp_cvx_103_011(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
     @testset "nlp_cvx_103_012" begin
         MINLPTests.nlp_cvx_103_012(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
                                    termination_target = MOI.OPTIMAL)
@@ -49,6 +74,16 @@ end
                                    termination_target = MOI.OPTIMAL)
     end
 
+    @testset "nlp_cvx_103_014" begin
+        MINLPTests.nlp_cvx_103_014(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
+    @testset "nlp_cvx_104_010" begin
+        MINLPTests.nlp_cvx_104_010(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
     @testset "nlp_cvx_105_010" begin
         MINLPTests.nlp_cvx_105_010(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
                                    termination_target = MOI.OPTIMAL)
@@ -79,6 +114,16 @@ end
                                    termination_target = MOI.OPTIMAL)
     end
 
+    @testset "nlp_cvx_202_010" begin
+        MINLPTests.nlp_cvx_202_010(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
+    @testset "nlp_cvx_202_011" begin
+        MINLPTests.nlp_cvx_202_011(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
     @testset "nlp_cvx_202_012" begin
         MINLPTests.nlp_cvx_202_012(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
                                    termination_target = MOI.OPTIMAL)
@@ -104,6 +149,11 @@ end
                                    termination_target = MOI.OPTIMAL)
     end
 
+    @testset "nlp_cvx_205_010" begin
+        MINLPTests.nlp_cvx_205_010(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
+                                   termination_target = MOI.OPTIMAL)
+    end
+
     # Wrong solution?!
     # @testset "nlp_cvx_206_010" begin
     #     MINLPTests.nlp_cvx_206_010(OPTIMIZER, OBJTOL, PRIMALTOL, DUALTOL;
diff --git a/test/MOI_additional.jl b/test/MOI_additional.jl
index ffec23a9..8bec3ab4 100644
--- a/test/MOI_additional.jl
+++ b/test/MOI_additional.jl
@@ -366,3 +366,25 @@ end
     # fail to delete them in wrong order
     @test_throws ErrorException MOI.delete(optimizer, x)
 end
+
+@testset "single variable objective" begin
+    optimizer = SCIP.Optimizer()
+
+    # Happy Path: add objective and retrieve it.
+    x = MOI.add_variable(optimizer)
+    obj = MOI.SingleVariable(x)
+    MOI.set(optimizer, MOI.ObjectiveFunction{MOI.SingleVariable}(), obj)
+    MOI.set(optimizer, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+    @test MOI.get(optimizer, MOI.ObjectiveFunction{MOI.SingleVariable}()) == obj
+    @test MOI.get(optimizer, MOI.ObjectiveSense()) == MOI.MAX_SENSE
+
+    MOI.empty!(optimizer)
+
+    # Error with type mismatch
+    x = MOI.add_variable(optimizer)
+    y = MOI.add_variable(optimizer)
+    aff_obj = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 3.0)
+    MOI.set(optimizer, MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(), aff_obj)
+    MOI.set(optimizer, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+    @test_throws ErrorException MOI.get(optimizer, MOI.ObjectiveFunction{MOI.SingleVariable}())
+end