diff --git a/ixmp4/core/optimization/data.py b/ixmp4/core/optimization/data.py
index c604dbe1..d3f8cbfb 100644
--- a/ixmp4/core/optimization/data.py
+++ b/ixmp4/core/optimization/data.py
@@ -29,3 +29,9 @@ def __init__(self, run: Run, **kwargs: Backend) -> None:
         self.scalars = ScalarRepository(_backend=self.backend, _run=run)
         self.tables = TableRepository(_backend=self.backend, _run=run)
         self.variables = VariableRepository(_backend=self.backend, _run=run)
+
+    def remove_solution(self) -> None:
+        for equation in self.equations.list():
+            equation.remove_data()
+        for variable in self.variables.list():
+            variable.remove_data()
diff --git a/ixmp4/core/optimization/equation.py b/ixmp4/core/optimization/equation.py
index 58efbd18..09c49b1a 100644
--- a/ixmp4/core/optimization/equation.py
+++ b/ixmp4/core/optimization/equation.py
@@ -40,7 +40,7 @@ def data(self) -> dict[str, Any]:
         return self._model.data
 
     def add(self, data: dict[str, Any] | pd.DataFrame) -> None:
-        """Adds data to an existing Equation."""
+        """Adds data to the Equation."""
         self.backend.optimization.equations.add_data(
             equation_id=self._model.id, data=data
         )
@@ -49,7 +49,7 @@ def add(self, data: dict[str, Any] | pd.DataFrame) -> None:
         )
 
     def remove_data(self) -> None:
-        """Removes data from an existing Equation."""
+        """Removes all data from the Equation."""
         self.backend.optimization.equations.remove_data(equation_id=self._model.id)
         self._model = self.backend.optimization.equations.get(
             run_id=self._model.run__id, name=self._model.name
diff --git a/ixmp4/core/optimization/parameter.py b/ixmp4/core/optimization/parameter.py
index 9d423632..ce8bf4ec 100644
--- a/ixmp4/core/optimization/parameter.py
+++ b/ixmp4/core/optimization/parameter.py
@@ -40,7 +40,7 @@ def data(self) -> dict[str, Any]:
         return self._model.data
 
     def add(self, data: dict[str, Any] | pd.DataFrame) -> None:
-        """Adds data to an existing Parameter."""
+        """Adds data to the Parameter."""
         self.backend.optimization.parameters.add_data(
             parameter_id=self._model.id, data=data
         )
diff --git a/ixmp4/core/optimization/table.py b/ixmp4/core/optimization/table.py
index 3c536ec5..4869b51b 100644
--- a/ixmp4/core/optimization/table.py
+++ b/ixmp4/core/optimization/table.py
@@ -40,7 +40,7 @@ def data(self) -> dict[str, Any]:
         return self._model.data
 
     def add(self, data: dict[str, Any] | pd.DataFrame) -> None:
-        """Adds data to an existing Table."""
+        """Adds data to the Table."""
         self.backend.optimization.tables.add_data(table_id=self._model.id, data=data)
         self._model = self.backend.optimization.tables.get(
             run_id=self._model.run__id, name=self._model.name
diff --git a/ixmp4/core/optimization/variable.py b/ixmp4/core/optimization/variable.py
index c66b7792..0daf3359 100644
--- a/ixmp4/core/optimization/variable.py
+++ b/ixmp4/core/optimization/variable.py
@@ -40,7 +40,7 @@ def data(self) -> dict[str, Any]:
         return self._model.data
 
     def add(self, data: dict[str, Any] | pd.DataFrame) -> None:
-        """Adds data to an existing Variable."""
+        """Adds data to the Variable."""
         self.backend.optimization.variables.add_data(
             variable_id=self._model.id, data=data
         )
@@ -49,7 +49,7 @@ def add(self, data: dict[str, Any] | pd.DataFrame) -> None:
         )
 
     def remove_data(self) -> None:
-        """Removes data from an existing Variable."""
+        """Removes all data from the Variable."""
         self.backend.optimization.variables.remove_data(variable_id=self._model.id)
         self._model = self.backend.optimization.variables.get(
             run_id=self._model.run__id, name=self._model.name
diff --git a/ixmp4/data/db/optimization/equation/model.py b/ixmp4/data/db/optimization/equation/model.py
index e7588043..5c589b63 100644
--- a/ixmp4/data/db/optimization/equation/model.py
+++ b/ixmp4/data/db/optimization/equation/model.py
@@ -26,16 +26,17 @@ class Equation(base.BaseModel):
 
     @validates("data")
     def validate_data(self, key: Any, data: dict[str, Any]) -> dict[str, Any]:
-        if data == {}:
+        if not bool(data):
             return data
         data_to_validate = copy.deepcopy(data)
         del data_to_validate["levels"]
         del data_to_validate["marginals"]
-        _ = utils.validate_data(
-            host=self,
-            data=data_to_validate,
-            columns=self.columns,
-        )
+        if bool(data_to_validate):
+            _ = utils.validate_data(
+                host=self,
+                data=data_to_validate,
+                columns=self.columns,
+            )
         return data
 
     __table_args__ = (db.UniqueConstraint("name", "run__id"),)
diff --git a/ixmp4/data/db/optimization/equation/repository.py b/ixmp4/data/db/optimization/equation/repository.py
index 34f9fdcd..393c5ae8 100644
--- a/ixmp4/data/db/optimization/equation/repository.py
+++ b/ixmp4/data/db/optimization/equation/repository.py
@@ -79,11 +79,7 @@ def _add_column(  # type: ignore[no-untyped-def]
             **kwargs,
         )
 
-    def add(
-        self,
-        run_id: int,
-        name: str,
-    ) -> Equation:
+    def add(self, run_id: int, name: str) -> Equation:
         equation = Equation(name=name, run__id=run_id)
         equation.set_creation_info(auth_context=self.backend.auth_context)
         self.session.add(equation)
diff --git a/ixmp4/data/db/optimization/indexset/repository.py b/ixmp4/data/db/optimization/indexset/repository.py
index c559ce23..69e5eac8 100644
--- a/ixmp4/data/db/optimization/indexset/repository.py
+++ b/ixmp4/data/db/optimization/indexset/repository.py
@@ -95,5 +95,4 @@ def add_data(
             Literal["float", "int", "str"], type(_data[0]).__name__
         )
 
-        self.session.add(indexset)
         self.session.commit()
diff --git a/ixmp4/data/db/optimization/scalar/repository.py b/ixmp4/data/db/optimization/scalar/repository.py
index 3ad02f51..e79af22d 100644
--- a/ixmp4/data/db/optimization/scalar/repository.py
+++ b/ixmp4/data/db/optimization/scalar/repository.py
@@ -89,8 +89,7 @@ def update(
 
         self.session.execute(exc)
         self.session.commit()
-        scalar: Scalar = self.get_by_id(id)
-        return scalar
+        return self.get_by_id(id)
 
     @guard("view")
     def list(self, **kwargs: Unpack[EnumerateKwargs]) -> Iterable[Scalar]:
diff --git a/ixmp4/data/db/optimization/table/repository.py b/ixmp4/data/db/optimization/table/repository.py
index 27dbef2c..dcdad5be 100644
--- a/ixmp4/data/db/optimization/table/repository.py
+++ b/ixmp4/data/db/optimization/table/repository.py
@@ -154,5 +154,4 @@ def add_data(self, table_id: int, data: dict[str, Any] | pd.DataFrame) -> None:
             orient="list"
         )  # type: ignore[assignment]
 
-        self.session.add(table)
         self.session.commit()
diff --git a/ixmp4/data/db/optimization/utils.py b/ixmp4/data/db/optimization/utils.py
index 3ff3bd8c..d612b4f3 100644
--- a/ixmp4/data/db/optimization/utils.py
+++ b/ixmp4/data/db/optimization/utils.py
@@ -10,13 +10,10 @@
 
 def collect_indexsets_to_check(
     columns: list["Column"],
-) -> dict[str, Any]:
+) -> dict[str, list[float] | list[int] | list[str]]:
     """Creates a {key:value} dict from linked Column.names and their
     IndexSet.data."""
-    collection: dict[str, Any] = {}
-    for column in columns:
-        collection[column.name] = column.indexset.data
-    return collection
+    return {column.name: column.indexset.data for column in columns}
 
 
 def validate_data(
diff --git a/ixmp4/data/db/optimization/variable/model.py b/ixmp4/data/db/optimization/variable/model.py
index 32cfae59..bae48e2e 100644
--- a/ixmp4/data/db/optimization/variable/model.py
+++ b/ixmp4/data/db/optimization/variable/model.py
@@ -27,16 +27,17 @@ class OptimizationVariable(base.BaseModel):
 
     @validates("data")
     def validate_data(self, key: Any, data: dict[str, Any]) -> dict[str, Any]:
-        if data == {}:
+        if not bool(data):
             return data
         data_to_validate = copy.deepcopy(data)
         del data_to_validate["levels"]
         del data_to_validate["marginals"]
-        _ = utils.validate_data(
-            host=self,
-            data=data_to_validate,
-            columns=self.columns,
-        )
+        if bool(data_to_validate):
+            _ = utils.validate_data(
+                host=self,
+                data=data_to_validate,
+                columns=self.columns,
+            )
         return data
 
     __table_args__ = (db.UniqueConstraint("name", "run__id"),)
diff --git a/ixmp4/data/db/optimization/variable/repository.py b/ixmp4/data/db/optimization/variable/repository.py
index 76aa13f3..c8bed49e 100644
--- a/ixmp4/data/db/optimization/variable/repository.py
+++ b/ixmp4/data/db/optimization/variable/repository.py
@@ -178,11 +178,14 @@ def add_data(self, variable_id: int, data: dict[str, Any] | pd.DataFrame) -> Non
 
         index_list = [column.name for column in variable.columns]
         existing_data = pd.DataFrame(variable.data)
-        if not existing_data.empty:
-            existing_data.set_index(index_list, inplace=True)
-        variable.data = (
-            data.set_index(index_list).combine_first(existing_data).reset_index()
-        ).to_dict(orient="list")  # type: ignore[assignment]
+        if index_list:
+            data = data.set_index(index_list)
+            if not existing_data.empty:
+                existing_data.set_index(index_list, inplace=True)
+        data = data.combine_first(existing_data)
+        if index_list:
+            data = data.reset_index()
+        variable.data = data.to_dict(orient="list")  # type: ignore[assignment]
 
         self.session.commit()
 
diff --git a/tests/core/test_optimization_variable.py b/tests/core/test_optimization_variable.py
index 183e65f5..0dec1d45 100644
--- a/tests/core/test_optimization_variable.py
+++ b/tests/core/test_optimization_variable.py
@@ -273,6 +273,14 @@ def test_variable_add_data(self, platform: ixmp4.Platform) -> None:
         )
         assert_unordered_equality(expected, pd.DataFrame(variable_4.data))
 
+        # Test adding to scalar variable raises
+        with pytest.raises(
+            OptimizationDataValidationError,
+            match="Trying to add data to unknown Columns!",
+        ):
+            variable_5 = run.optimization.variables.create("Variable 5")
+            variable_5.add(data={"foo": ["bar"], "levels": [1], "marginals": [0]})
+
     def test_variable_remove_data(self, platform: ixmp4.Platform) -> None:
         run = platform.runs.create("Model", "Scenario")
         indexset = run.optimization.indexsets.create("Indexset")
diff --git a/tests/core/test_run.py b/tests/core/test_run.py
index e75aa4f5..1a128555 100644
--- a/tests/core/test_run.py
+++ b/tests/core/test_run.py
@@ -181,3 +181,29 @@ def delete_all_datapoints(self, run: ixmp4.Run) -> None:
             run.iamc.remove(cat, type=ixmp4.DataPoint.Type.CATEGORICAL)
         if not datetime.empty:
             run.iamc.remove(datetime, type=ixmp4.DataPoint.Type.DATETIME)
+
+    def test_run_remove_solution(self, platform: ixmp4.Platform) -> None:
+        run = platform.runs.create("Model", "Scenario")
+        indexset = run.optimization.indexsets.create("Indexset")
+        indexset.add(["foo", "bar"])
+        test_data = {
+            "Indexset": ["bar", "foo"],
+            "levels": [2.0, 1],
+            "marginals": [0, "test"],
+        }
+        run.optimization.equations.create(
+            "Equation",
+            constrained_to_indexsets=[indexset.name],
+        ).add(test_data)
+        run.optimization.variables.create(
+            "Variable",
+            constrained_to_indexsets=[indexset.name],
+        ).add(test_data)
+
+        run.optimization.remove_solution()
+        # Need to fetch them here even if fetched before because API layer might not
+        # forward changes automatically
+        equation = run.optimization.equations.get("Equation")
+        variable = run.optimization.variables.get("Variable")
+        assert equation.data == {}
+        assert variable.data == {}
diff --git a/tests/data/test_optimization_variable.py b/tests/data/test_optimization_variable.py
index f7969134..92bbb4c9 100644
--- a/tests/data/test_optimization_variable.py
+++ b/tests/data/test_optimization_variable.py
@@ -83,6 +83,18 @@ def test_create_variable(self, platform: ixmp4.Platform) -> None:
                 column_names=["Dimension 1"],
             )
 
+        # Test that giving column_names, but not constrained_to_indexsets raises
+        with pytest.raises(
+            OptimizationItemUsageError,
+            match="Received `column_names` to name columns, but no "
+            "`constrained_to_indexsets`",
+        ):
+            _ = platform.backend.optimization.variables.create(
+                run_id=run.id,
+                name="Variable 0",
+                column_names=["Dimension 1"],
+            )
+
         # Test mismatch in constrained_to_indexsets and column_names raises
         with pytest.raises(OptimizationItemUsageError, match="not equal in length"):
             _ = platform.backend.optimization.variables.create(