diff --git a/src/tranquilo/robustify_subproblem_solution.py b/src/tranquilo/robustify_subproblem_solution.py
new file mode 100644
index 0000000..4f0f6e7
--- /dev/null
+++ b/src/tranquilo/robustify_subproblem_solution.py
@@ -0,0 +1,91 @@
+import numpy as np
+import estimagic as em
+import functools
+
+
+def robustify_subproblem_sphere_solver(solver):
+    @functools.wraps(solver)
+    def wrapped_solver(model, x_candidate, lower_bounds, upper_bounds, **kwargs):
+        try:
+            result = solver(
+                model=model,
+                x_candidate=x_candidate,
+                lower_bounds=lower_bounds,
+                upper_bounds=upper_bounds,
+                **kwargs,
+            )
+        except Exception:
+            result = _robust_sphere_solver(
+                model=model,
+                x_candidate=x_candidate,
+            )
+        return result
+
+    return wrapped_solver
+
+
+def _robust_sphere_solver(model, x_candidate):
+    """Robustly find the minimizer of a quadratic model on the unit sphere.
+
+    Args:
+        model (ScalarModel): The fitted model of which we want to find the minimum.
+        x_candidate (np.ndarray): The candidate solution.
+
+    Returns:
+        dict: The result of the solver.
+        - x (np.ndarray): The minimizer.
+        - criterion (float): The value of the criterion at the minimizer.
+        - n_iterations (int): The number of iterations.
+        - success (bool): Whether the solver was successful.
+
+    """
+    criterion, derivative = _get_criterion_and_derivative(model)
+
+    # Run an unconstrained solver in the unit cube
+    # ==================================================================================
+    lower_bounds = -np.ones(len(x_candidate))
+    upper_bounds = np.ones(len(x_candidate))
+
+    res = em.minimize(
+        criterion=criterion,
+        params=x_candidate,
+        algorithm="scipy_lbfgsb",
+        derivative=derivative,
+        algo_options={
+            "stopping.max_iterations": len(x_candidate),
+        },
+        lower_bounds=lower_bounds,
+        upper_bounds=upper_bounds,
+    )
+
+    # Project the solution onto the unit sphere if necessary
+    # ==================================================================================
+    solution_lies_inside_sphere = np.linalg.norm(res.params) <= 1
+
+    if solution_lies_inside_sphere:
+        _minimizer = res.params
+        _criterion = res.criterion
+    else:
+        _minimizer = _project_onto_unit_sphere(res.params)
+        _criterion = criterion(_minimizer)
+
+    return {
+        "x": _minimizer,
+        "criterion": _criterion,
+        "n_iterations": res.n_iterations,
+        "success": True,
+    }
+
+
+def _get_criterion_and_derivative(model):
+    def criterion(x):
+        return model.predict(x)
+
+    def derivative(x):
+        return model.linear_terms + 2 * model.square_terms @ x
+
+    return criterion, derivative
+
+
+def _project_onto_unit_sphere(x):
+    return x / np.linalg.norm(x)
diff --git a/src/tranquilo/solve_subproblem.py b/src/tranquilo/solve_subproblem.py
index 645417b..2cab20e 100644
--- a/src/tranquilo/solve_subproblem.py
+++ b/src/tranquilo/solve_subproblem.py
@@ -19,6 +19,7 @@
     solve_multistart,
 )
 from tranquilo.options import SubsolverOptions
+from tranquilo.robustify_subproblem_solution import robustify_subproblem_sphere_solver
 
 
 def get_subsolver(sphere_solver, cube_solver, user_options=None):
@@ -87,6 +88,7 @@ def get_subsolver(sphere_solver, cube_solver, user_options=None):
         user_options=user_options,
         mandatory_signature=["model", "x_candidate"],
     )
+    _robust_sphere_solver = robustify_subproblem_sphere_solver(_sphere_subsolver)
 
     _cube_subsolver = get_component(
         name_or_func=cube_solver,
@@ -99,7 +101,7 @@ def get_subsolver(sphere_solver, cube_solver, user_options=None):
 
     solver = partial(
         _solve_subproblem_template,
-        sphere_solver=_sphere_subsolver,
+        sphere_solver=_robust_sphere_solver,
         cube_solver=_cube_subsolver,
     )