From 998cc8f58866ea93fa47ef342ebb3e2662dd1555 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Tue, 16 Aug 2022 13:03:21 +0200
Subject: [PATCH 01/12] Include bounds.

---
 src/estimagic/optimization/bhhh.py | 25 +++++++---
 tests/optimization/test_bhhh.py    | 80 +++++++++++++++++++++++-------
 2 files changed, 80 insertions(+), 25 deletions(-)

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 6e09412bd..655bbb7a3 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -33,6 +33,8 @@ def bhhh(
 def bhhh_internal(
     criterion_and_derivative,
     x,
+    lower_bounds,
+    upper_bounds,
     convergence_absolute_gradient_tolerance,
     stopping_max_iterations,
 ):
@@ -66,17 +68,19 @@ def bhhh_internal(
     initial_step_size = 1
     step_size = initial_step_size
 
-    niter = 1
-    while niter < stopping_max_iterations:
-        niter += 1
+    n_iter = 1
+    while n_iter < stopping_max_iterations:
+        n_iter += 1
 
         x_candidate = x_accepted + step_size * direction
+        x_candidate = _apply_bounds_to_x_candidate(
+            x_candidate, lower_bounds, upper_bounds
+        )
         criterion_candidate, gradient = criterion_and_derivative(x_candidate)
 
         # If previous step was accepted
         if step_size == initial_step_size:
             hessian_approx = np.dot(gradient.T, gradient)
-
         else:
             criterion_candidate, gradient = criterion_and_derivative(x_candidate)
 
@@ -85,11 +89,9 @@ def bhhh_internal(
             step_size /= 2
 
             if step_size <= 0.01:
-                # Accept step
                 x_accepted = x_candidate
                 criterion_accepted = criterion_candidate
 
-                # Reset step size
                 step_size = initial_step_size
 
         # If decrease in likelihood, calculate new direction vector
@@ -106,7 +108,6 @@ def bhhh_internal(
                 hessian_approx = np.dot(gradient.T, gradient)
                 direction = np.linalg.solve(hessian_approx, gradient_sum)
 
-            # Reset stepsize
             step_size = initial_step_size
 
         if gtol < convergence_absolute_gradient_tolerance:
@@ -115,8 +116,16 @@ def bhhh_internal(
     result_dict = {
         "solution_x": x_accepted,
         "solution_criterion": criterion_accepted,
-        "n_iterations": niter,
+        "n_iterations": n_iter,
         "message": "Under develpment",
     }
 
     return result_dict
+
+
+def _apply_bounds_to_x_candidate(x, lower_bounds, upper_bounds, bound_tol=0):
+    """Apply upper and lower bounds to the candidate vector."""
+    x = np.where(x <= lower_bounds + bound_tol, lower_bounds, x)
+    x = np.where(x >= upper_bounds - bound_tol, upper_bounds, x)
+
+    return x
diff --git a/tests/optimization/test_bhhh.py b/tests/optimization/test_bhhh.py
index 20098635c..67669166c 100644
--- a/tests/optimization/test_bhhh.py
+++ b/tests/optimization/test_bhhh.py
@@ -87,7 +87,6 @@ def criterion_and_derivative_probit(x):
 
     Returns:
         tuple: first entry is the criterion, second entry is the score
-
     """
     endog, exog = generate_test_data()
 
@@ -97,39 +96,86 @@ def criterion_and_derivative_probit(x):
     return -loglike(x), score(x)
 
 
+# =====================================================================================
+# Fixtures
+# =====================================================================================
+
+
 @pytest.fixture
-def result_statsmodels_logit():
+def result_logit_unbounded():
     endog, exog = generate_test_data()
-    result = sm.Logit(endog, exog).fit()
+    result_unbounded = sm.Logit(endog, exog).fit()
 
-    return result
+    return result_unbounded.params
 
 
 @pytest.fixture
-def result_statsmodels_probit():
+def result_probit_unbounded():
     endog, exog = generate_test_data()
-    result = sm.Probit(endog, exog).fit()
+    result_unbounded = sm.Probit(endog, exog).fit()
+
+    return result_unbounded.params
+
 
-    return result
+@pytest.fixture
+def result_logit_bounded():
+    return np.array([-5.0, -3.93455447, 6.49171767])
+
+
+@pytest.fixture
+def result_probit_bounded():
+    return np.array([-5.0, -2.56431715, 4.18468707])
+
+
+# =====================================================================================
+# Tests
+# =====================================================================================
+TEST_CASES = [
+    (
+        criterion_and_derivative_logit,
+        np.array([-np.inf, -np.inf, -np.inf]),
+        np.array([np.inf, np.inf, np.inf]),
+        "result_logit_unbounded",
+    ),
+    (
+        criterion_and_derivative_probit,
+        np.array([-np.inf, -np.inf, -np.inf]),
+        np.array([np.inf, np.inf, np.inf]),
+        "result_probit_unbounded",
+    ),
+    (
+        criterion_and_derivative_logit,
+        np.array([-5, -10, -10]),
+        np.array([10, 10, 10]),
+        "result_logit_bounded",
+    ),
+    (
+        criterion_and_derivative_probit,
+        np.array([-5, -10, -10]),
+        np.array([10, 10, 10]),
+        "result_probit_bounded",
+    ),
+]
 
 
 @pytest.mark.parametrize(
-    "criterion_and_derivative, result_statsmodels",
-    [
-        (criterion_and_derivative_logit, "result_statsmodels_logit"),
-        (criterion_and_derivative_probit, "result_statsmodels_probit"),
-    ],
+    "criterion_and_derivative, lower_bounds, upper_bounds, result_statsmodels",
+    TEST_CASES,
 )
-def test_maximum_likelihood(criterion_and_derivative, result_statsmodels, request):
-    result_expected = request.getfixturevalue(result_statsmodels)
+def test_maximum_likelihood(
+    criterion_and_derivative, lower_bounds, upper_bounds, result_statsmodels, request
+):
+    params_expected = request.getfixturevalue(result_statsmodels)
 
-    x = np.zeros(3)
+    x0 = np.zeros(3)
 
     result_bhhh = bhhh_internal(
         criterion_and_derivative,
-        x=x,
+        x=x0,
+        lower_bounds=lower_bounds,
+        upper_bounds=upper_bounds,
         convergence_absolute_gradient_tolerance=1e-8,
         stopping_max_iterations=200,
     )
 
-    aaae(result_bhhh["solution_x"], result_expected.params, decimal=4)
+    aaae(result_bhhh["solution_x"], params_expected, decimal=4)

From 3eeb5dd2b32c19839ff0d65bca7d2325180fb776 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Tue, 16 Aug 2022 14:23:32 +0200
Subject: [PATCH 02/12] Refactor.

---
 src/estimagic/optimization/bhhh.py | 149 +++++++++++++++++++++--------
 1 file changed, 110 insertions(+), 39 deletions(-)

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 655bbb7a3..bea83217c 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -42,7 +42,11 @@ def bhhh_internal(
 
     Args:
         criterion_and_derivative (callable): The objective function to be minimized.
-        x (np.ndarray): Initial guess of the parameter vector (starting points).
+        x (np.ndarray): Initial guess of the parameter vector x (starting points).
+        lower_bounds (np.ndarray): 1d array of shape (n,) with lower bounds
+            for the parameter vector x.
+        upper_bounds (np.ndarray): 1d array of shape (n,) with upper bounds
+            for the parameter vector x.
         convergence_absolute_gradient_tolerance (float): Stopping criterion for the
             gradient tolerance.
         stopping_max_iterations (int): Maximum number of iterations. If reached,
@@ -57,56 +61,52 @@ def bhhh_internal(
             solution vector or reaching stopping_max_iterations.
         - message (str): Message to the user. Currently it says: "Under development."
     """
-    criterion_accepted, gradient = criterion_and_derivative(x)
+    criterion_accepted, gradient_candidate = criterion_and_derivative(x)
     x_accepted = x
 
-    hessian_approx = np.dot(gradient.T, gradient)
-    gradient_sum = np.sum(gradient, axis=0)
-    direction = np.linalg.solve(hessian_approx, gradient_sum)
-    gtol = np.dot(gradient_sum, direction)
+    hessian_approx = gradient_candidate.T @ gradient_candidate
+    direction, gtol = _calculate_new_direction_vector(
+        gradient_candidate, hessian_approx
+    )
 
     initial_step_size = 1
     step_size = initial_step_size
 
-    n_iter = 1
-    while n_iter < stopping_max_iterations:
-        n_iter += 1
-
-        x_candidate = x_accepted + step_size * direction
-        x_candidate = _apply_bounds_to_x_candidate(
-            x_candidate, lower_bounds, upper_bounds
+    for _n_iter in range(stopping_max_iterations):
+        (
+            x_candidate,
+            criterion_candidate,
+            gradient_candidate,
+            hessian_approx,
+        ) = find_new_candidates(
+            x_accepted,
+            direction,
+            hessian_approx,
+            lower_bounds,
+            upper_bounds,
+            step_size,
+            initial_step_size,
+            criterion_and_derivative,
         )
-        criterion_candidate, gradient = criterion_and_derivative(x_candidate)
-
-        # If previous step was accepted
-        if step_size == initial_step_size:
-            hessian_approx = np.dot(gradient.T, gradient)
-        else:
-            criterion_candidate, gradient = criterion_and_derivative(x_candidate)
 
-        # Line search
         if np.sum(criterion_candidate) > np.sum(criterion_accepted):
-            step_size /= 2
+            x_accepted, criterion_accepted, step_size = determine_new_step_size(
+                x_accepted,
+                criterion_accepted,
+                x_candidate,
+                criterion_candidate,
+                step_size,
+                initial_step_size,
+            )
 
-            if step_size <= 0.01:
-                x_accepted = x_candidate
-                criterion_accepted = criterion_candidate
-
-                step_size = initial_step_size
-
-        # If decrease in likelihood, calculate new direction vector
         else:
-            # Accept step
             x_accepted = x_candidate
             criterion_accepted = criterion_candidate
 
-            gradient_sum = np.sum(gradient, axis=0)
-            direction = np.linalg.solve(hessian_approx, gradient_sum)
-            gtol = np.dot(gradient_sum, direction)
-
-            if gtol < 0:
-                hessian_approx = np.dot(gradient.T, gradient)
-                direction = np.linalg.solve(hessian_approx, gradient_sum)
+            direction, hessian_approx, gtol = determine_new_search_direction(
+                gradient_candidate,
+                hessian_approx,
+            )
 
             step_size = initial_step_size
 
@@ -116,13 +116,84 @@ def bhhh_internal(
     result_dict = {
         "solution_x": x_accepted,
         "solution_criterion": criterion_accepted,
-        "n_iterations": n_iter,
-        "message": "Under develpment",
+        "n_iterations": _n_iter,
+        "message": "Under development",
     }
 
     return result_dict
 
 
+def find_new_candidates(
+    x_accepted,
+    direction,
+    hessian_approx,
+    lower_bounds,
+    upper_bounds,
+    step_size,
+    initial_step_size,
+    criterion_and_derivative,
+):
+    """Find new candidates for x, criterion, gradient, and hessian."""
+    x_candidate = x_accepted + step_size * direction
+    x_candidate = _apply_bounds_to_x_candidate(x_candidate, lower_bounds, upper_bounds)
+
+    criterion_candidate, gradient_candidate = criterion_and_derivative(x_candidate)
+
+    if step_size == initial_step_size:
+        hessian_approx = gradient_candidate.T @ gradient_candidate
+    else:
+        criterion_candidate, gradient_candidate = criterion_and_derivative(x_candidate)
+
+    return x_candidate, criterion_candidate, gradient_candidate, hessian_approx
+
+
+def determine_new_step_size(
+    x_accepted,
+    criterion_accepted,
+    x_candidate,
+    criterion_candidate,
+    step_size,
+    initial_step_size,
+):
+    """Determine new step size and accept candidates."""
+    step_size /= 2
+
+    if step_size <= 0.01:
+        x_accepted = x_candidate
+        criterion_accepted = criterion_candidate
+
+        step_size = initial_step_size
+
+    return x_accepted, criterion_accepted, step_size
+
+
+def determine_new_search_direction(
+    gradient_candidate,
+    hessian_approx,
+):
+    """Determine new search direction and accept candidates."""
+    direction, gtol = _calculate_new_direction_vector(
+        gradient_candidate, hessian_approx
+    )
+
+    if gtol < 0:
+        hessian_approx = gradient_candidate.T @ gradient_candidate
+        direction, _ = _calculate_new_direction_vector(
+            gradient_candidate, hessian_approx
+        )
+
+    return direction, hessian_approx, gtol
+
+
+def _calculate_new_direction_vector(gradient_candidate, hessian_approx):
+    """Calculate new direction vector."""
+    gradient_sum = np.sum(gradient_candidate, axis=0)
+    direction = np.linalg.solve(hessian_approx, gradient_sum)
+    gtol = gradient_sum @ direction
+
+    return direction, gtol
+
+
 def _apply_bounds_to_x_candidate(x, lower_bounds, upper_bounds, bound_tol=0):
     """Apply upper and lower bounds to the candidate vector."""
     x = np.where(x <= lower_bounds + bound_tol, lower_bounds, x)

From 11a4db29e03b2963cdd7761c9af9565a858161cf Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Wed, 24 Aug 2022 18:43:07 +0200
Subject: [PATCH 03/12] Implement bounds a la lbfgsb.

---
 docs/source/algorithms.rst         |   4 +-
 src/estimagic/optimization/bhhh.py | 224 +++++++++++++++--------------
 tests/optimization/test_bhhh.py    |  25 ++--
 3 files changed, 127 insertions(+), 126 deletions(-)

diff --git a/docs/source/algorithms.rst b/docs/source/algorithms.rst
index d2f1aeb3f..fd99739c7 100644
--- a/docs/source/algorithms.rst
+++ b/docs/source/algorithms.rst
@@ -559,14 +559,14 @@ We implement a few algorithms from scratch. They are currently considered experi
 
         "bhhh"
 
-    Minimize a likelihood function using the BHHH algorithm.
+    Minimize a likelihood function using the bounded BHHH algorithm.
 
     BHHH (:cite:`Berndt1974`) can - and should ONLY - be used for minimizing
     (or maximizing) a likelihood. It is similar to the Newton-Raphson
     algorithm, but replaces the Hessian matrix with the outer product of the
     gradient. This approximation is based on the information matrix equality
     (:cite:`Halbert1982`) and is thus only vaid when minimizing (or maximizing)
-    a likelihood.
+    a likelihood. Bounds, i.e. box constraints, are supported.
 
     The criterion function :func:`func` should return a dictionary with
     at least the entry ``{"contributions": array_or_pytree}`` where ``array_or_pytree``
diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index bea83217c..28cdae1fe 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -1,4 +1,4 @@
-"""Implement Berndt-Hall-Hall-Hausman (BHHH) algorithm."""
+"""Implement the Berndt-Hall-Hall-Hausman (BHHH) algorithm."""
 import numpy as np
 from estimagic.decorators import mark_minimizer
 
@@ -12,18 +12,24 @@
 def bhhh(
     criterion_and_derivative,
     x,
+    lower_bounds,
+    upper_bounds,
     *,
     convergence_absolute_gradient_tolerance=1e-8,
+    convergence_relative_gradient_tolerance=1e-8,
     stopping_max_iterations=200,
 ):
-    """Minimize a likelihood function using the BHHH algorithm.
+    """Minimize a likelihood function using the bounded BHHH algorithm.
 
     For details, see :ref:`_own_algorithms`.
     """
     result_dict = bhhh_internal(
         criterion_and_derivative,
         x=x,
+        lower_bounds=lower_bounds,
+        upper_bounds=upper_bounds,
         convergence_absolute_gradient_tolerance=convergence_absolute_gradient_tolerance,
+        convergence_relative_gradient_tolerance=convergence_relative_gradient_tolerance,
         stopping_max_iterations=stopping_max_iterations,
     )
 
@@ -36,167 +42,171 @@ def bhhh_internal(
     lower_bounds,
     upper_bounds,
     convergence_absolute_gradient_tolerance,
+    convergence_relative_gradient_tolerance,
     stopping_max_iterations,
 ):
-    """Minimize a likelihood function using the BHHH algorithm.
+    """Minimize a likelihood function using the bounded BHHH algorithm.
 
     Args:
-        criterion_and_derivative (callable): The objective function to be minimized.
+        criterion_and_derivative (callable): A function returning the tuple:
+            - criterion (np.ndarray): Likelihood contributions of shape (n_obs,)
+            - derivative (np.ndarray): Jacobian matrix of shape (n_obs, n_params)
         x (np.ndarray): Initial guess of the parameter vector x (starting points).
-        lower_bounds (np.ndarray): 1d array of shape (n,) with lower bounds
-            for the parameter vector x.
-        upper_bounds (np.ndarray): 1d array of shape (n,) with upper bounds
+        lower_bounds (np.ndarray): 1d array of shape (n_params,) with lower bounds
             for the parameter vector x.
+        upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
+            for the parameter vector x
         convergence_absolute_gradient_tolerance (float): Stopping criterion for the
-            gradient tolerance.
+            absolute gradient tolerance.
+        convergence_relative_gradient_tolerance (float): Stopping criterion for the
+            relative gradient tolerance.
         stopping_max_iterations (int): Maximum number of iterations. If reached,
             terminate.
 
     Returns:
         (dict) Result dictionary containing:
 
-        - solution_x (np.ndarray): Solution vector of shape (n,).
-        - solution_criterion (np.ndarray): Likelihood at the solution. Shape (n_obs,).
+        - solution_x (np.ndarray): Solution vector of shape (n_params,).
+        - solution_criterion (np.ndarray): Likelihood contributions at the solution.
+            Shape (n_obs,).
+        - solution_loglikelihood (float): Value of the log-likelihood at the solution.
         - n_iterations (int): Number of iterations the algorithm ran before finding a
             solution vector or reaching stopping_max_iterations.
         - message (str): Message to the user. Currently it says: "Under development."
     """
-    criterion_accepted, gradient_candidate = criterion_and_derivative(x)
+    tol = {
+        "abs": convergence_absolute_gradient_tolerance,
+        "rel": convergence_relative_gradient_tolerance,
+    }
+    n_params = len(x)
+
     x_accepted = x
+    criterion_accepted, jacobian = criterion_and_derivative(x_accepted)
+    gradient = np.sum(jacobian, axis=0)
 
-    hessian_approx = gradient_candidate.T @ gradient_candidate
-    direction, gtol = _calculate_new_direction_vector(
-        gradient_candidate, hessian_approx
+    norm_pg0 = np.linalg.norm(
+        x_accepted - np.clip(x_accepted - gradient, lower_bounds, upper_bounds)
+    )
+    norm_pg = norm_pg0
+
+    inactive_set = estimate_epsilon_inactive_set(
+        x_accepted, norm_pg, lower_bounds, upper_bounds
+    )
+
+    gradient_reduced = gradient[inactive_set]
+    hessian_reduced = jacobian[:, inactive_set].T @ jacobian[:, inactive_set]
+    direction = determine_descent_direction(
+        gradient, gradient_reduced, hessian_reduced, inactive_set, n_params
     )
 
     initial_step_size = 1
     step_size = initial_step_size
 
     for _n_iter in range(stopping_max_iterations):
-        (
-            x_candidate,
-            criterion_candidate,
-            gradient_candidate,
-            hessian_approx,
-        ) = find_new_candidates(
-            x_accepted,
-            direction,
-            hessian_approx,
-            lower_bounds,
-            upper_bounds,
-            step_size,
-            initial_step_size,
-            criterion_and_derivative,
+
+        x_candidate = np.clip(
+            x_accepted + step_size * direction, lower_bounds, upper_bounds
         )
 
+        criterion_candidate, jacobian = criterion_and_derivative(x_candidate)
+
+        if step_size == initial_step_size:
+            hessian_reduced = jacobian[:, inactive_set].T @ jacobian[:, inactive_set]
+
         if np.sum(criterion_candidate) > np.sum(criterion_accepted):
-            x_accepted, criterion_accepted, step_size = determine_new_step_size(
-                x_accepted,
-                criterion_accepted,
-                x_candidate,
-                criterion_candidate,
-                step_size,
-                initial_step_size,
-            )
+            step_size /= 2
 
+            if step_size <= 0.01:
+                x_accepted = x_candidate
+                criterion_accepted = criterion_candidate
+
+                step_size = initial_step_size
         else:
             x_accepted = x_candidate
             criterion_accepted = criterion_candidate
 
-            direction, hessian_approx, gtol = determine_new_search_direction(
-                gradient_candidate,
-                hessian_approx,
+            gradient = np.sum(jacobian, axis=0)
+            gradient_reduced = gradient[inactive_set]
+
+            direction = determine_descent_direction(
+                gradient,
+                gradient_reduced,
+                hessian_reduced,
+                inactive_set,
+                n_params,
             )
 
+            norm_pg = np.linalg.norm(
+                x_accepted - np.clip(x_accepted - gradient, lower_bounds, upper_bounds)
+            )
+            inactive_set = estimate_epsilon_inactive_set(
+                x_accepted, norm_pg, lower_bounds, upper_bounds
+            )
+
+            if norm_pg < 0:
+                gradient_reduced = gradient[inactive_set]
+                hessian_reduced = (
+                    jacobian[:, inactive_set].T @ jacobian[:, inactive_set]
+                )
+
+                direction = determine_descent_direction(
+                    gradient,
+                    gradient_reduced,
+                    hessian_reduced,
+                    inactive_set,
+                    n_params,
+                )
+
             step_size = initial_step_size
 
-        if gtol < convergence_absolute_gradient_tolerance:
+        if norm_pg < tol["abs"] + tol["rel"] * norm_pg0:
             break
 
     result_dict = {
         "solution_x": x_accepted,
         "solution_criterion": criterion_accepted,
+        "solution_loglikelihood": np.sum(criterion_accepted),
         "n_iterations": _n_iter,
-        "message": "Under development",
+        "message": "Under develpment",
     }
 
     return result_dict
 
 
-def find_new_candidates(
-    x_accepted,
-    direction,
-    hessian_approx,
-    lower_bounds,
-    upper_bounds,
-    step_size,
-    initial_step_size,
-    criterion_and_derivative,
-):
-    """Find new candidates for x, criterion, gradient, and hessian."""
-    x_candidate = x_accepted + step_size * direction
-    x_candidate = _apply_bounds_to_x_candidate(x_candidate, lower_bounds, upper_bounds)
-
-    criterion_candidate, gradient_candidate = criterion_and_derivative(x_candidate)
-
-    if step_size == initial_step_size:
-        hessian_approx = gradient_candidate.T @ gradient_candidate
-    else:
-        criterion_candidate, gradient_candidate = criterion_and_derivative(x_candidate)
-
-    return x_candidate, criterion_candidate, gradient_candidate, hessian_approx
-
+def estimate_epsilon_inactive_set(x, norm_gradient, lower_bounds, upper_bounds):
+    """Estimate the set of epsilon-inactive bound constraints up to a tolerance.
 
-def determine_new_step_size(
-    x_accepted,
-    criterion_accepted,
-    x_candidate,
-    criterion_candidate,
-    step_size,
-    initial_step_size,
-):
-    """Determine new step size and accept candidates."""
-    step_size /= 2
+    The set of epsilon-inactive indices underestimates (overestimates) the actual
+    set of inactive (active) indices.
 
-    if step_size <= 0.01:
-        x_accepted = x_candidate
-        criterion_accepted = criterion_candidate
+        x (np.ndarray): Current candidate vector.
+        norm_gradient (float): Norm of the projected gradient.
+        lower_bounds (np.ndarray): 1d array of shape (n_params,) with lower bounds
+            for the parameter vector x.
+        upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
+            for the parameter vector x
+    """
+    epsilon = min(np.min(upper_bounds - lower_bounds) / 2, norm_gradient)
 
-        step_size = initial_step_size
+    inactive_set = np.where(
+        (lower_bounds + epsilon < x) & (x < upper_bounds - epsilon)
+    )[0]
 
-    return x_accepted, criterion_accepted, step_size
+    return inactive_set
 
 
-def determine_new_search_direction(
-    gradient_candidate,
-    hessian_approx,
+def determine_descent_direction(
+    gradient_candidate, gradient_reduced, hessian_reduced, inactive_set, n_params
 ):
-    """Determine new search direction and accept candidates."""
-    direction, gtol = _calculate_new_direction_vector(
-        gradient_candidate, hessian_approx
-    )
-
-    if gtol < 0:
-        hessian_approx = gradient_candidate.T @ gradient_candidate
-        direction, _ = _calculate_new_direction_vector(
-            gradient_candidate, hessian_approx
-        )
-
-    return direction, hessian_approx, gtol
-
-
-def _calculate_new_direction_vector(gradient_candidate, hessian_approx):
-    """Calculate new direction vector."""
-    gradient_sum = np.sum(gradient_candidate, axis=0)
-    direction = np.linalg.solve(hessian_approx, gradient_sum)
-    gtol = gradient_sum @ direction
+    """Determine the new descent serach direction."""
+    direction = np.linalg.solve(hessian_reduced, gradient_reduced)
 
-    return direction, gtol
+    direction_active = gradient_candidate.copy()
+    direction_active[inactive_set] = 0
 
+    direction_projected = np.zeros(n_params)
+    direction_projected[inactive_set] = direction
 
-def _apply_bounds_to_x_candidate(x, lower_bounds, upper_bounds, bound_tol=0):
-    """Apply upper and lower bounds to the candidate vector."""
-    x = np.where(x <= lower_bounds + bound_tol, lower_bounds, x)
-    x = np.where(x >= upper_bounds - bound_tol, upper_bounds, x)
+    direction_all = direction_active + direction_projected
 
-    return x
+    return direction_all
diff --git a/tests/optimization/test_bhhh.py b/tests/optimization/test_bhhh.py
index 67669166c..8375714a5 100644
--- a/tests/optimization/test_bhhh.py
+++ b/tests/optimization/test_bhhh.py
@@ -13,7 +13,7 @@
 def generate_test_data():
     rng = get_rng(seed=12)
 
-    num_observations = 5000
+    num_observations = 5_000
     x1 = rng.multivariate_normal([0, 0], [[1, 0.75], [0.75, 1]], num_observations)
     x2 = rng.multivariate_normal([1, 4], [[1, 0.75], [0.75, 1]], num_observations)
 
@@ -104,7 +104,7 @@ def criterion_and_derivative_probit(x):
 @pytest.fixture
 def result_logit_unbounded():
     endog, exog = generate_test_data()
-    result_unbounded = sm.Logit(endog, exog).fit()
+    result_unbounded = sm.Logit(endog, exog).fit(disp=True)
 
     return result_unbounded.params
 
@@ -112,24 +112,20 @@ def result_logit_unbounded():
 @pytest.fixture
 def result_probit_unbounded():
     endog, exog = generate_test_data()
-    result_unbounded = sm.Probit(endog, exog).fit()
+    result_unbounded = sm.Probit(endog, exog).fit(disp=True)
 
     return result_unbounded.params
 
 
 @pytest.fixture
 def result_logit_bounded():
-    return np.array([-5.0, -3.93455447, 6.49171767])
-
-
-@pytest.fixture
-def result_probit_bounded():
-    return np.array([-5.0, -2.56431715, 4.18468707])
+    return np.array([-1.866, -1.5223, 2.7153])
 
 
 # =====================================================================================
 # Tests
 # =====================================================================================
+
 TEST_CASES = [
     (
         criterion_and_derivative_logit,
@@ -145,16 +141,10 @@ def result_probit_bounded():
     ),
     (
         criterion_and_derivative_logit,
-        np.array([-5, -10, -10]),
-        np.array([10, 10, 10]),
+        np.array([-5, -1_000, -1_000]),
+        np.array([1_000, 1_000, 1_000]),
         "result_logit_bounded",
     ),
-    (
-        criterion_and_derivative_probit,
-        np.array([-5, -10, -10]),
-        np.array([10, 10, 10]),
-        "result_probit_bounded",
-    ),
 ]
 
 
@@ -175,6 +165,7 @@ def test_maximum_likelihood(
         lower_bounds=lower_bounds,
         upper_bounds=upper_bounds,
         convergence_absolute_gradient_tolerance=1e-8,
+        convergence_relative_gradient_tolerance=1e-8,
         stopping_max_iterations=200,
     )
 

From 41daf40fd8eae9377eef43eec35a34f1027e6889 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Mon, 29 Aug 2022 17:41:58 +0200
Subject: [PATCH 04/12] Implement alternative algorithm for constrained case.

---
 src/estimagic/optimization/bhhh.py            |  193 +-
 .../fixtures/telco_churn_clean.csv            | 7033 +++++++++++++++++
 tests/optimization/test_bhhh.py               |  266 +-
 3 files changed, 7443 insertions(+), 49 deletions(-)
 create mode 100644 tests/optimization/fixtures/telco_churn_clean.csv

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 28cdae1fe..7f6ddfaee 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -19,24 +19,35 @@ def bhhh(
     convergence_relative_gradient_tolerance=1e-8,
     stopping_max_iterations=200,
 ):
-    """Minimize a likelihood function using the bounded BHHH algorithm.
+    """Minimize a likelihood function using the box-constrained BHHH algorithm.
 
     For details, see :ref:`_own_algorithms`.
     """
-    result_dict = bhhh_internal(
-        criterion_and_derivative,
-        x=x,
-        lower_bounds=lower_bounds,
-        upper_bounds=upper_bounds,
-        convergence_absolute_gradient_tolerance=convergence_absolute_gradient_tolerance,
-        convergence_relative_gradient_tolerance=convergence_relative_gradient_tolerance,
-        stopping_max_iterations=stopping_max_iterations,
-    )
+    if np.isinf(lower_bounds).all() and np.isinf(upper_bounds).all():
+        result_dict = bhhh_unconstrained(
+            criterion_and_derivative,
+            x,
+            lower_bounds,
+            upper_bounds,
+            convergence_absolute_gradient_tolerance,
+            convergence_relative_gradient_tolerance,
+            stopping_max_iterations,
+        )
+    else:
+        result_dict = bhhh_box_constrained(
+            criterion_and_derivative,
+            x,
+            lower_bounds,
+            upper_bounds,
+            convergence_absolute_gradient_tolerance,
+            convergence_relative_gradient_tolerance,
+            stopping_max_iterations,
+        )
 
     return result_dict
 
 
-def bhhh_internal(
+def bhhh_unconstrained(
     criterion_and_derivative,
     x,
     lower_bounds,
@@ -94,7 +105,9 @@ def bhhh_internal(
     )
 
     gradient_reduced = gradient[inactive_set]
-    hessian_reduced = jacobian[:, inactive_set].T @ jacobian[:, inactive_set]
+    hessian_approx = jacobian.T @ jacobian
+    hessian_reduced = hessian_approx[inactive_set[:, np.newaxis], inactive_set]
+
     direction = determine_descent_direction(
         gradient, gradient_reduced, hessian_reduced, inactive_set, n_params
     )
@@ -111,7 +124,8 @@ def bhhh_internal(
         criterion_candidate, jacobian = criterion_and_derivative(x_candidate)
 
         if step_size == initial_step_size:
-            hessian_reduced = jacobian[:, inactive_set].T @ jacobian[:, inactive_set]
+            hessian_approx = jacobian.T @ jacobian
+            hessian_reduced = hessian_approx[inactive_set[:, np.newaxis], inactive_set]
 
         if np.sum(criterion_candidate) > np.sum(criterion_accepted):
             step_size /= 2
@@ -145,9 +159,10 @@ def bhhh_internal(
 
             if norm_pg < 0:
                 gradient_reduced = gradient[inactive_set]
-                hessian_reduced = (
-                    jacobian[:, inactive_set].T @ jacobian[:, inactive_set]
-                )
+                hessian_approx = jacobian.T @ jacobian
+                hessian_reduced = hessian_approx[
+                    inactive_set[:, np.newaxis], inactive_set
+                ]
 
                 direction = determine_descent_direction(
                     gradient,
@@ -198,7 +213,7 @@ def estimate_epsilon_inactive_set(x, norm_gradient, lower_bounds, upper_bounds):
 def determine_descent_direction(
     gradient_candidate, gradient_reduced, hessian_reduced, inactive_set, n_params
 ):
-    """Determine the new descent serach direction."""
+    """Determine the new descent (search) direction."""
     direction = np.linalg.solve(hessian_reduced, gradient_reduced)
 
     direction_active = gradient_candidate.copy()
@@ -210,3 +225,147 @@ def determine_descent_direction(
     direction_all = direction_active + direction_projected
 
     return direction_all
+
+
+def bhhh_box_constrained(
+    criterion_and_derivative,
+    x,
+    lower_bounds,
+    upper_bounds,
+    convergence_absolute_gradient_tolerance,
+    stopping_max_iterations,
+):
+    """Minimize a likelihood function using the box-constrained BHHH algorithm.
+
+    Args:
+        criterion_and_derivative (callable): A function returning the tuple:
+            - criterion (np.ndarray): Likelihood contributions of shape (n_obs,)
+            - derivative (np.ndarray): Jacobian matrix of shape (n_obs, n_params)
+        x (np.ndarray): Initial guess of the parameter vector x (starting points).
+        lower_bounds (np.ndarray): 1d array of shape (n_params,) with lower bounds
+            for the parameter vector x.
+        upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
+            for the parameter vector x
+        convergence_absolute_gradient_tolerance (float): Stopping criterion for the
+            absolute gradient tolerance.
+        convergence_relative_gradient_tolerance (float): Stopping criterion for the
+            relative gradient tolerance.
+        stopping_max_iterations (int): Maximum number of iterations. If reached,
+            terminate.
+
+    Returns:
+        (dict) Result dictionary containing:
+
+        - solution_x (np.ndarray): Solution vector of shape (n_params,).
+        - solution_criterion (np.ndarray): Likelihood contributions at the solution.
+            Shape (n_obs,).
+        - solution_loglikelihood (float): Value of the log-likelihood at the solution.
+        - n_iterations (int): Number of iterations the algorithm ran before finding a
+            solution vector or reaching stopping_max_iterations.
+        - message (str): Message to the user. Currently it says: "Under development."
+    """
+    _zero_threshold = np.finfo(float).eps
+    n_params = len(x)
+    step_min = 1e-6
+
+    critic_limit = convergence_absolute_gradient_tolerance
+
+    crit = 1
+    step_len_optimal = 1
+
+    jacobian = criterion_and_derivative(x, task="derivative")
+    gradient = np.sum(jacobian, axis=0)
+
+    norm_pg = np.linalg.norm(x - np.clip(x - gradient, lower_bounds, upper_bounds))
+    inactive_set = estimate_epsilon_inactive_set(x, norm_pg, lower_bounds, upper_bounds)
+
+    for _n_iter in range(stopping_max_iterations):
+        jacobian = criterion_and_derivative(x, task="derivative")
+        gradient = np.sum(jacobian, axis=0)
+
+        gradient_reduced = gradient[inactive_set]
+        hessian_approx = jacobian.T @ jacobian
+        hessian_reduced = hessian_approx[inactive_set[:, np.newaxis], inactive_set]
+
+        direction_projected = determine_descent_direction(
+            gradient, gradient_reduced, hessian_reduced, inactive_set, n_params
+        )
+
+        step_len_optimal = find_optimal_step_len(
+            x,
+            direction_projected,
+            lower_bounds,
+            upper_bounds,
+            step_min,
+            criterion_and_derivative,
+        )
+
+        x_candidate = np.clip(
+            x + step_len_optimal * direction_projected, lower_bounds, upper_bounds
+        )
+
+        crit = np.max(
+            np.abs(((x_candidate - x) + _zero_threshold) / (x + _zero_threshold))
+            / step_len_optimal
+        )
+
+        x = x_candidate
+
+        if crit < critic_limit:
+            break
+
+        norm_pg = np.linalg.norm(x - np.clip(x - gradient, lower_bounds, upper_bounds))
+        inactive_set = estimate_epsilon_inactive_set(
+            x, norm_pg, lower_bounds, upper_bounds
+        )
+
+    solution_criterion = criterion_and_derivative(x, task="criterion")
+
+    result_dict = {
+        "solution_x": x,
+        "solution_criterion": solution_criterion,
+        "solution_loglikelihood": np.sum(solution_criterion),
+        "n_iterations": _n_iter,
+        "message": "Under develpment",
+    }
+
+    return result_dict
+
+
+def find_optimal_step_len(
+    x,
+    direction_projected,
+    lower_bounds,
+    upper_bounds,
+    step_min,
+    criterion_and_derivative,
+):
+    """Find optimal step length."""
+    step_len_trial = 2
+    step_len_optimal = 1
+
+    loglike_higher = 1
+    loglike_lower = 0
+
+    while (loglike_higher > loglike_lower) & (step_len_trial >= step_min):
+        step_len_trial = step_len_trial / 2
+
+        criterion_candidate_higher = criterion_and_derivative(
+            np.clip(
+                x + step_len_trial * direction_projected, lower_bounds, upper_bounds
+            ),
+            task="criterion",
+        )
+        criterion_candidate_lower = criterion_and_derivative(
+            np.clip(
+                x + step_len_trial * direction_projected / 2, lower_bounds, upper_bounds
+            ),
+            task="criterion",
+        )
+
+        loglike_higher = np.sum(criterion_candidate_higher)
+        loglike_lower = np.sum(criterion_candidate_lower)
+
+        step_len_optimal = step_len_trial
+
+    return step_len_optimal
diff --git a/tests/optimization/fixtures/telco_churn_clean.csv b/tests/optimization/fixtures/telco_churn_clean.csv
new file mode 100644
index 000000000..cc1fd3b1e
--- /dev/null
+++ b/tests/optimization/fixtures/telco_churn_clean.csv
@@ -0,0 +1,7033 @@
+gender,SeniorCitizen,Partner,Dependents,tenure,PhoneService,PaperlessBilling,MonthlyCharges,Churn
+0,0,1,0,1,0,1,29.85,0
+1,0,0,0,34,1,0,56.95,0
+1,0,0,0,2,1,1,53.85,1
+1,0,0,0,45,0,0,42.3,0
+0,0,0,0,2,1,1,70.7,1
+0,0,0,0,8,1,1,99.65,1
+1,0,0,1,22,1,1,89.1,0
+0,0,0,0,10,0,0,29.75,0
+0,0,1,0,28,1,1,104.8,1
+1,0,0,1,62,1,0,56.15,0
+1,0,1,1,13,1,1,49.95,0
+1,0,0,0,16,1,0,18.95,0
+1,0,1,0,58,1,0,100.35,0
+1,0,0,0,49,1,1,103.7,1
+1,0,0,0,25,1,1,105.5,0
+0,0,1,1,69,1,0,113.25,0
+0,0,0,0,52,1,0,20.65,0
+1,0,0,1,71,1,0,106.7,0
+0,0,1,1,10,1,0,55.2,1
+0,0,0,0,21,1,1,90.05,0
+1,1,0,0,1,0,1,39.65,1
+1,0,1,0,12,1,0,19.8,0
+1,0,0,0,1,1,0,20.15,1
+0,0,1,0,58,1,1,59.9,0
+1,0,1,1,49,1,0,59.6,0
+0,0,0,0,30,1,1,55.3,0
+1,0,1,1,47,1,1,99.35,1
+1,0,1,1,1,0,0,30.2,1
+1,0,1,0,72,1,1,90.25,0
+0,0,0,1,17,1,1,64.7,1
+0,1,1,0,71,1,1,96.35,0
+1,1,1,0,2,1,1,95.5,0
+0,0,1,1,27,1,0,66.15,0
+1,0,0,0,1,1,0,20.2,0
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+0,0,0,0,54,1,0,111.1,1
+1,0,0,0,2,1,0,20.0,0
+0,0,0,0,42,1,1,67.7,0
+0,0,1,1,58,1,0,25.15,0
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+0,0,1,1,25,1,1,89.1,1
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+0,0,0,0,37,1,1,101.9,1
+0,0,1,1,14,1,1,91.65,1
+0,1,0,0,4,1,1,88.85,1
+0,0,1,0,48,1,1,60.6,0
+0,1,0,0,3,0,0,25.3,1
+0,0,1,0,8,1,0,65.5,0
+1,0,0,0,1,1,1,95.45,1
+1,0,0,0,67,1,0,19.95,0
+1,0,0,0,13,1,1,91.1,1
+0,0,1,1,45,1,1,54.15,1
+1,0,0,0,49,1,1,74.6,0
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+0,0,1,0,63,1,1,81.15,0
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+0,0,1,0,31,1,1,49.2,0
+0,0,1,0,64,1,0,19.45,0
+0,0,1,0,62,1,1,104.3,0
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+0,0,0,0,6,1,1,89.5,1
+0,0,1,0,21,1,1,86.05,0
+0,0,1,1,72,1,0,25.2,0
+0,0,1,1,32,0,1,35.15,0
+0,1,0,0,71,1,1,99.65,0
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+1,0,1,0,3,0,1,35.15,1
+1,0,0,1,12,1,1,73.75,1
+0,1,0,0,8,1,0,101.35,1
+0,0,1,1,35,1,0,24.3,0
+0,0,1,1,3,1,1,80.7,0
+1,0,0,0,3,1,1,89.85,0
+1,0,0,0,53,1,1,61.1,0
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+0,0,1,1,48,1,1,99.7,0
+0,0,1,1,6,1,0,55.9,1
+1,1,1,0,3,1,1,105.9,1
+1,0,1,1,54,0,0,46.0,0
+0,0,0,0,1,1,0,43.95,1
+1,0,0,0,62,1,1,80.4,0
+1,0,1,1,22,1,1,100.05,0
+1,0,0,0,1,1,1,45.1,1
+1,1,0,0,51,1,1,94.0,0
+0,0,0,0,30,1,1,68.95,0
+0,1,1,0,56,1,1,68.45,0
+1,0,1,0,35,1,1,69.0,0
+0,1,0,0,64,0,1,43.85,0
+1,0,0,1,30,0,1,44.5,0
+0,0,1,1,25,1,1,18.7,0
+0,0,0,0,41,1,0,70.25,1
+1,0,0,1,9,1,0,55.35,1
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+1,0,0,1,70,1,1,114.6,0
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+1,0,0,0,9,1,1,85.5,0
+0,0,1,1,69,1,1,108.75,0
+1,1,1,0,43,1,1,103.0,1
+0,0,1,0,72,1,1,97.85,0
+1,0,0,1,44,1,0,19.55,0
+0,0,1,0,72,1,1,84.05,0
+0,0,1,0,33,1,1,103.75,1
+1,0,0,0,54,1,0,89.4,0
+1,0,0,1,27,1,0,19.7,0
+1,0,0,0,54,1,1,79.85,0
+1,0,0,0,3,1,0,74.45,0
+0,0,1,0,53,1,0,74.1,0
+1,0,0,0,1,1,1,69.35,1
+1,0,1,0,15,1,0,18.8,0
+0,0,0,0,56,1,1,73.85,1
+1,0,1,1,5,1,0,64.4,0
+1,0,1,1,48,1,1,55.8,0
+0,0,0,1,25,1,0,20.05,0
+0,0,1,0,3,1,1,75.15,1
+1,0,0,0,58,1,0,99.15,0
+1,0,0,0,10,1,1,56.75,0
+1,0,0,0,1,1,1,69.6,1
+1,0,1,0,71,1,1,104.15,0
+0,0,1,1,65,1,1,110.8,0
+1,0,1,1,5,1,1,80.15,1
+0,0,0,0,28,0,1,35.75,0
+0,0,1,1,67,1,1,69.9,0
+1,0,0,0,35,1,1,89.2,0
+0,0,1,1,72,0,0,55.65,0
+0,0,1,1,61,0,1,50.7,0
+1,1,0,0,68,1,1,20.0,0
+0,0,1,1,1,0,0,30.5,1
+1,0,0,0,3,1,1,19.1,0
+1,0,1,1,70,1,0,98.3,1
+0,0,0,1,48,0,0,45.55,0
+1,0,1,1,68,1,1,101.05,0
+0,0,0,0,47,1,1,103.7,0
+1,0,1,1,32,0,1,36.25,0
+0,0,1,1,5,1,1,49.4,0
+1,0,0,0,49,1,1,19.9,0
+0,0,0,0,48,1,1,107.4,1
+0,1,0,0,13,1,1,82.0,1
+1,0,0,0,15,1,1,19.8,0
+0,0,0,0,12,1,1,45.05,0
+1,1,0,0,67,1,1,64.55,0
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+0,0,1,1,13,1,1,19.75,0
+0,0,0,0,38,1,1,89.1,0
+0,1,1,0,42,1,1,95.55,0
+0,0,1,0,24,1,1,75.4,1
+0,1,1,0,27,1,1,101.25,1
+1,0,1,1,9,1,1,102.6,0
+0,0,1,0,49,0,1,56.3,0
+0,1,1,0,61,1,1,94.2,0
+1,0,0,1,50,0,1,43.05,0
+1,1,0,0,25,1,1,89.5,1
+0,1,0,0,22,1,1,74.4,1
+1,0,0,0,1,1,0,20.5,1
+1,0,0,0,4,1,1,74.35,1
+0,1,1,0,18,1,0,99.75,1
+0,1,0,0,56,1,1,111.95,1
+0,1,1,0,53,1,1,94.0,0
+0,0,1,0,51,1,1,98.85,0
+0,0,1,1,24,1,0,64.35,0
+0,0,1,0,62,1,1,72.0,0
+1,0,0,0,24,1,1,49.7,0
+0,0,1,1,70,1,0,80.7,0
+1,0,0,1,1,0,0,24.2,0
+1,1,0,0,16,0,1,39.0,1
+0,0,0,0,8,1,1,65.45,0
+0,0,1,1,72,1,1,74.35,0
+0,0,0,0,23,1,0,83.2,0
+1,0,0,1,31,0,0,25.0,0
+1,0,1,1,37,0,1,40.2,0
+0,0,1,0,30,1,1,94.1,1
+0,1,0,0,35,1,1,108.35,0
+0,0,1,1,23,1,1,69.5,0
+0,0,1,1,20,1,0,76.0,0
+1,0,1,0,36,1,1,93.6,0
+0,0,0,0,8,1,0,95.65,1
+1,0,1,1,71,1,0,100.55,0
+0,1,1,0,50,1,1,88.05,1
+0,0,1,1,43,1,1,24.45,0
+1,0,0,1,57,1,0,89.55,0
+0,0,1,1,41,1,1,66.5,1
+0,0,1,0,27,1,1,76.1,0
+0,0,0,0,13,1,1,80.5,0
+1,0,0,0,3,0,1,35.45,1
+0,0,0,0,67,1,0,20.55,0
+1,0,0,0,3,1,1,49.9,1
+0,0,1,0,64,1,1,105.4,0
+1,0,0,0,26,0,0,35.75,0
+1,0,0,0,38,1,1,95.1,0
+1,0,1,0,23,1,0,19.3,0
+1,0,0,0,40,1,1,104.5,1
+1,1,1,0,72,0,1,63.1,0
+0,1,0,0,3,1,1,75.05,1
+1,0,0,0,23,1,1,81.0,1
+0,1,1,0,1,1,1,74.45,1
+0,0,0,0,4,1,1,60.4,1
+0,0,1,0,62,1,1,84.95,0
+0,0,0,0,40,1,1,93.4,0
+1,0,0,0,41,1,1,89.2,0
+1,1,1,0,34,1,1,85.2,0
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+1,0,1,1,12,1,1,19.2,0
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+1,0,0,0,72,1,1,104.95,0
+0,1,1,0,63,1,1,103.5,0
+1,0,1,0,44,1,1,84.8,0
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+0,0,1,0,68,1,0,64.1,0
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+0,0,0,0,2,1,1,20.05,0
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+1,1,0,0,1,1,1,75.75,1
+1,0,0,0,38,1,1,69.5,0
+0,0,0,0,67,1,1,102.95,1
+1,0,0,0,19,1,1,78.7,0
+0,0,0,0,12,0,0,60.65,0
+0,0,0,0,72,1,1,21.15,0
+1,0,1,1,24,1,1,84.8,0
+0,0,1,1,72,1,1,103.2,0
+0,0,1,1,11,0,1,29.6,0
+1,1,1,0,4,1,1,74.4,1
+1,0,0,0,66,1,1,105.65,0
diff --git a/tests/optimization/test_bhhh.py b/tests/optimization/test_bhhh.py
index 8375714a5..b2af0bc14 100644
--- a/tests/optimization/test_bhhh.py
+++ b/tests/optimization/test_bhhh.py
@@ -2,9 +2,12 @@
 from functools import partial
 
 import numpy as np
+import pandas as pd
 import pytest
 import statsmodels.api as sm
-from estimagic.optimization.bhhh import bhhh_internal
+from estimagic.config import TEST_FIXTURES_DIR
+from estimagic.optimization.bhhh import bhhh_box_constrained
+from estimagic.optimization.bhhh import bhhh_unconstrained
 from estimagic.utilities import get_rng
 from numpy.testing import assert_array_almost_equal as aaae
 from scipy.stats import norm
@@ -27,6 +30,20 @@ def generate_test_data():
     return endog, exog
 
 
+def ibm_test_data():
+    df = pd.read_csv(TEST_FIXTURES_DIR / "telco_churn_clean.csv")
+
+    exog = df.drop(columns="Churn").to_numpy()
+    intercept = np.ones((exog.shape[0], 1))
+    exog = np.hstack((intercept, exog))
+    endog = df[["Churn"]].to_numpy()
+    endog = endog.reshape(
+        len(endog),
+    )
+
+    return endog, exog
+
+
 def _cdf_logit(x):
     return 1 / (1 + np.exp(-x))
 
@@ -62,38 +79,132 @@ def get_score_probit(endog, exog, x):
     return derivative_loglikelihood[:, None] * exog
 
 
-def criterion_and_derivative_logit(x):
+def criterion_and_derivative_logit(x, task="criterion_and_derivative"):
     """Return Logit criterion and derivative.
 
     Args:
         x (np.ndarray): Parameter vector of shape (n_obs,).
+        task (str): If task=="criterion", compute log-likelihood.
+            If task=="derivative", compute derivative.
+            If task="criterion_and_derivative", compute both.
 
     Returns:
-        tuple: first entry is the criterion, second entry is the score
-
+        (np.ndarray or tuple): If task=="criterion" it returns the output of
+            criterion, which is a 1d numpy array.
+            If task=="derivative" it returns the first derivative of criterion,
+            which is 2d numpy array.
+            If task=="criterion_and_derivative" it returns both as a tuple.
     """
     endog, exog = generate_test_data()
-    score = partial(get_score_logit, endog, exog)
-    loglike = partial(get_loglikelihood_logit, endog, exog)
+    result = ()
+
+    if "criterion" in task:
+        loglike = partial(get_loglikelihood_logit, endog, exog)
+        result += (-loglike(x),)
+    if "derivative" in task:
+        score = partial(get_score_logit, endog, exog)
+        result += (score(x),)
+
+    if len(result) == 1:
+        (result,) = result
+
+    return result
+
+
+def criterion_and_derivative_logit_ibm(x, task="criterion_and_derivative"):
+    """Return Logit criterion and derivative.
+
+    Args:
+        x (np.ndarray): Parameter vector of shape (n_obs,).
+        task (str): If task=="criterion", compute log-likelihood.
+            If task=="derivative", compute derivative.
+            If task="criterion_and_derivative", compute both.
+
+    Returns:
+        (np.ndarray or tuple): If task=="criterion" it returns the output of
+            criterion, which is a 1d numpy array.
+            If task=="derivative" it returns the first derivative of criterion,
+            which is 2d numpy array.
+            If task=="criterion_and_derivative" it returns both as a tuple.
+    """
+    endog, exog = ibm_test_data()
+    result = ()
 
-    return -loglike(x), score(x)
+    if "criterion" in task:
+        loglike = partial(get_loglikelihood_logit, endog, exog)
+        result += (-loglike(x),)
+    if "derivative" in task:
+        score = partial(get_score_logit, endog, exog)
+        result += (score(x),)
 
+    if len(result) == 1:
+        (result,) = result
 
-def criterion_and_derivative_probit(x):
+    return result
+
+
+def criterion_and_derivative_probit(x, task="criterion_and_derivative"):
     """Return Probit criterion and derivative.
 
     Args:
         x (np.ndarray): Parameter vector of shape (n_obs,).
+        task (str): If task=="criterion", compute log-likelihood.
+            If task=="derivative", compute derivative.
+            If task="criterion_and_derivative", compute both.
 
     Returns:
-        tuple: first entry is the criterion, second entry is the score
+        (np.ndarray or tuple): If task=="criterion" it returns the output of
+            criterion, which is a 1d numpy array.
+            If task=="derivative" it returns the first derivative of criterion,
+            which is 2d numpy array.
+            If task=="criterion_and_derivative" it returns both as a tuple.
     """
     endog, exog = generate_test_data()
+    result = ()
+
+    if "criterion" in task:
+        loglike = partial(get_loglikelihood_probit, endog, exog)
+        result += (-loglike(x),)
+    if "derivative" in task:
+        score = partial(get_score_probit, endog, exog)
+        result += (score(x),)
+
+    if len(result) == 1:
+        (result,) = result
+
+    return result
+
+
+def criterion_and_derivative_probit_ibm(x, task="criterion_and_derivative"):
+    """Return Probit criterion and derivative.
+
+    Args:
+        x (np.ndarray): Parameter vector of shape (n_obs,).
+        task (str): If task=="criterion", compute log-likelihood.
+            If task=="derivative", compute derivative.
+            If task="criterion_and_derivative", compute both.
+
+    Returns:
+        (np.ndarray or tuple): If task=="criterion" it returns the output of
+            criterion, which is a 1d numpy array.
+            If task=="derivative" it returns the first derivative of criterion,
+            which is 2d numpy array.
+            If task=="criterion_and_derivative" it returns both as a tuple.
+    """
+    endog, exog = ibm_test_data()
+    result = ()
+
+    if "criterion" in task:
+        loglike = partial(get_loglikelihood_probit, endog, exog)
+        result += (-loglike(x),)
+    if "derivative" in task:
+        score = partial(get_score_probit, endog, exog)
+        result += (score(x),)
 
-    score = partial(get_score_probit, endog, exog)
-    loglike = partial(get_loglikelihood_probit, endog, exog)
+    if len(result) == 1:
+        (result,) = result
 
-    return -loglike(x), score(x)
+    return result
 
 
 # =====================================================================================
@@ -106,7 +217,7 @@ def result_logit_unbounded():
     endog, exog = generate_test_data()
     result_unbounded = sm.Logit(endog, exog).fit(disp=True)
 
-    return result_unbounded.params
+    return result_unbounded
 
 
 @pytest.fixture
@@ -114,52 +225,96 @@ def result_probit_unbounded():
     endog, exog = generate_test_data()
     result_unbounded = sm.Probit(endog, exog).fit(disp=True)
 
-    return result_unbounded.params
+    return result_unbounded
 
 
 @pytest.fixture
 def result_logit_bounded():
-    return np.array([-1.866, -1.5223, 2.7153])
+    endog, exog = generate_test_data()
+    result_bounded = sm.Logit(endog, exog).fit(
+        method="lbfgs", bounds=((-5, np.inf), (-10, 10), (-10, 10)), disp=False
+    )
+
+    return result_bounded
+
+
+@pytest.fixture
+def result_probit_bounded():
+    endog, exog = generate_test_data()
+    result_bounded = sm.Logit(endog, exog).fit(
+        method="lbfgs", bounds=((-5, np.inf), (-10, 10), (-10, 10)), disp=False
+    )
+
+    return result_bounded
+
+
+@pytest.fixture
+def result_logit_ibm():
+    endog, exog = ibm_test_data()
+    result_unbounded = sm.Logit(endog, exog).fit(disp=False)
+
+    return result_unbounded
+
+
+@pytest.fixture
+def result_probit_ibm():
+    endog, exog = ibm_test_data()
+    result_unbounded = sm.Probit(endog, exog).fit(disp=False)
+
+    return result_unbounded
 
 
 # =====================================================================================
 # Tests
 # =====================================================================================
 
-TEST_CASES = [
+TEST_CASES_UNBOUNDED = [
     (
         criterion_and_derivative_logit,
-        np.array([-np.inf, -np.inf, -np.inf]),
-        np.array([np.inf, np.inf, np.inf]),
+        np.zeros(3),
+        -np.ones(3) * np.inf,
+        np.ones(3) * np.inf,
         "result_logit_unbounded",
     ),
     (
         criterion_and_derivative_probit,
-        np.array([-np.inf, -np.inf, -np.inf]),
-        np.array([np.inf, np.inf, np.inf]),
+        np.zeros(3),
+        -np.ones(3) * np.inf,
+        np.ones(3) * np.inf,
         "result_probit_unbounded",
     ),
     (
-        criterion_and_derivative_logit,
-        np.array([-5, -1_000, -1_000]),
-        np.array([1_000, 1_000, 1_000]),
-        "result_logit_bounded",
+        criterion_and_derivative_logit_ibm,
+        np.zeros(9),
+        -np.ones(9) * np.inf,
+        np.ones(9) * np.inf,
+        "result_logit_ibm",
+    ),
+    (
+        criterion_and_derivative_probit_ibm,
+        np.zeros(9),
+        -np.ones(9) * np.inf,
+        np.ones(9) * np.inf,
+        "result_probit_ibm",
     ),
 ]
 
 
 @pytest.mark.parametrize(
-    "criterion_and_derivative, lower_bounds, upper_bounds, result_statsmodels",
-    TEST_CASES,
+    "criterion_and_derivative, x0, lower_bounds, upper_bounds, expected",
+    TEST_CASES_UNBOUNDED,
 )
 def test_maximum_likelihood(
-    criterion_and_derivative, lower_bounds, upper_bounds, result_statsmodels, request
+    criterion_and_derivative,
+    x0,
+    lower_bounds,
+    upper_bounds,
+    expected,
+    request,
 ):
-    params_expected = request.getfixturevalue(result_statsmodels)
-
-    x0 = np.zeros(3)
+    params_expected = request.getfixturevalue(expected)
 
-    result_bhhh = bhhh_internal(
+    result_bhhh = bhhh_unconstrained(
         criterion_and_derivative,
         x=x0,
         lower_bounds=lower_bounds,
@@ -169,4 +324,51 @@ def test_maximum_likelihood(
         stopping_max_iterations=200,
     )
 
-    aaae(result_bhhh["solution_x"], params_expected, decimal=4)
+    aaae(result_bhhh["solution_x"], params_expected.params, decimal=4)
+
+
+TEST_CASES_BOUNDED = [
+    (
+        criterion_and_derivative_logit,
+        np.zeros(3),
+        np.array([-5, -100, -100]),
+        np.array([100, 100, 100]),
+        "result_logit_bounded",
+        4,
+    ),
+    (
+        criterion_and_derivative_probit,
+        np.zeros(3),
+        np.array([-5, -np.inf, -np.inf]),
+        np.array([10, np.inf, np.inf]),
+        "result_probit_bounded",
+        0,
+    ),
+]
+
+
+@pytest.mark.parametrize(
+    "criterion_and_derivative, x0, lower_bounds, upper_bounds, expected, digits",
+    TEST_CASES_BOUNDED,
+)
+def test_maximum_likelihood_bounded(
+    criterion_and_derivative,
+    x0,
+    lower_bounds,
+    upper_bounds,
+    expected,
+    decimals,
+    request,
+):
+    params_expected = request.getfixturevalue(expected)
+
+    result_bhhh = bhhh_box_constrained(
+        criterion_and_derivative,
+        x=x0,
+        lower_bounds=lower_bounds,
+        upper_bounds=upper_bounds,
+        convergence_absolute_gradient_tolerance=1e-4,
+        stopping_max_iterations=200,
+    )
+
+    aaae(result_bhhh["solution_x"], params_expected.params, decimal=decimals)

From 4bfc0174beef5a64d6a3294e77400cb1179f0b0c Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Mon, 29 Aug 2022 20:32:27 +0200
Subject: [PATCH 05/12] Fix.

---
 src/estimagic/optimization/bhhh.py | 92 ++++++++++++++++--------------
 tests/optimization/test_bhhh.py    | 20 +++----
 2 files changed, 59 insertions(+), 53 deletions(-)

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 7f6ddfaee..5c463f481 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -188,45 +188,6 @@ def bhhh_unconstrained(
     return result_dict
 
 
-def estimate_epsilon_inactive_set(x, norm_gradient, lower_bounds, upper_bounds):
-    """Estimate the set of epsilon-inactive bound constraints up to a tolerance.
-
-    The set of epsilon-inactive indices underestimates (overestimates) the actual
-    set of inactive (active) indices.
-
-        x (np.ndarray): Current candidate vector.
-        norm_gradient (float): Norm of the projected gradient.
-        lower_bounds (np.ndarray): 1d array of shape (n_params,) with lower bounds
-            for the parameter vector x.
-        upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
-            for the parameter vector x
-    """
-    epsilon = min(np.min(upper_bounds - lower_bounds) / 2, norm_gradient)
-
-    inactive_set = np.where(
-        (lower_bounds + epsilon < x) & (x < upper_bounds - epsilon)
-    )[0]
-
-    return inactive_set
-
-
-def determine_descent_direction(
-    gradient_candidate, gradient_reduced, hessian_reduced, inactive_set, n_params
-):
-    """Determine the new descent (search) direction."""
-    direction = np.linalg.solve(hessian_reduced, gradient_reduced)
-
-    direction_active = gradient_candidate.copy()
-    direction_active[inactive_set] = 0
-
-    direction_projected = np.zeros(n_params)
-    direction_projected[inactive_set] = direction
-
-    direction_all = direction_active + direction_projected
-
-    return direction_all
-
-
 def bhhh_box_constrained(
     criterion_and_derivative,
     x,
@@ -276,8 +237,12 @@ def bhhh_box_constrained(
     jacobian = criterion_and_derivative(x, task="derivative")
     gradient = np.sum(jacobian, axis=0)
 
-    norm_pg = np.linalg.norm(x - np.clip(x - gradient, lower_bounds, upper_bounds))
-    inactive_set = estimate_epsilon_inactive_set(x, norm_pg, lower_bounds, upper_bounds)
+    norm_proj_grad = np.linalg.norm(
+        x - np.clip(x - gradient, lower_bounds, upper_bounds)
+    )
+    inactive_set = estimate_epsilon_inactive_set(
+        x, norm_proj_grad, lower_bounds, upper_bounds
+    )
 
     for _n_iter in range(stopping_max_iterations):
         jacobian = criterion_and_derivative(x, task="derivative")
@@ -314,9 +279,11 @@ def bhhh_box_constrained(
         if crit < critic_limit:
             break
 
-        norm_pg = np.linalg.norm(x - np.clip(x - gradient, lower_bounds, upper_bounds))
+        norm_proj_grad = np.linalg.norm(
+            x - np.clip(x - gradient, lower_bounds, upper_bounds)
+        )
         inactive_set = estimate_epsilon_inactive_set(
-            x, norm_pg, lower_bounds, upper_bounds
+            x, norm_proj_grad, lower_bounds, upper_bounds
         )
 
     solution_criterion = criterion_and_derivative(x, task="criterion")
@@ -332,6 +299,45 @@ def bhhh_box_constrained(
     return result_dict
 
 
+def estimate_epsilon_inactive_set(x, norm_gradient, lower_bounds, upper_bounds):
+    """Estimate the set of epsilon-inactive bound constraints up to a tolerance.
+
+    The set of epsilon-inactive indices underestimates (overestimates) the actual
+    set of inactive (active) indices.
+
+        x (np.ndarray): Current candidate vector.
+        norm_gradient (float): Norm of the projected gradient.
+        lower_bounds (np.ndarray): 1d array of shape (n_params,) with lower bounds
+            for the parameter vector x.
+        upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
+            for the parameter vector x
+    """
+    epsilon = min(np.min(upper_bounds - lower_bounds) / 2, norm_gradient)
+
+    inactive_set = np.where(
+        (lower_bounds + epsilon < x) & (x < upper_bounds - epsilon)
+    )[0]
+
+    return inactive_set
+
+
+def determine_descent_direction(
+    gradient_candidate, gradient_reduced, hessian_reduced, inactive_set, n_params
+):
+    """Determine the new descent (search) direction."""
+    direction = np.linalg.solve(hessian_reduced, gradient_reduced)
+
+    direction_active = gradient_candidate.copy()
+    direction_active[inactive_set] = 0
+
+    direction_projected = np.zeros(n_params)
+    direction_projected[inactive_set] = direction
+
+    direction_all = direction_active + direction_projected
+
+    return direction_all
+
+
 def find_optimal_step_len(
     x,
     direction_projected,
diff --git a/tests/optimization/test_bhhh.py b/tests/optimization/test_bhhh.py
index b2af0bc14..6260daa58 100644
--- a/tests/optimization/test_bhhh.py
+++ b/tests/optimization/test_bhhh.py
@@ -272,29 +272,29 @@ def result_probit_ibm():
     (
         criterion_and_derivative_logit,
         np.zeros(3),
-        -np.ones(3) * np.inf,
-        np.ones(3) * np.inf,
+        np.full(3, -np.inf),
+        np.full(3, np.inf),
         "result_logit_unbounded",
     ),
     (
         criterion_and_derivative_probit,
         np.zeros(3),
-        -np.ones(3) * np.inf,
-        np.ones(3) * np.inf,
+        np.full(3, -np.inf),
+        np.full(3, np.inf),
         "result_probit_unbounded",
     ),
     (
         criterion_and_derivative_logit_ibm,
         np.zeros(9),
-        -np.ones(9) * np.inf,
-        np.ones(9) * np.inf,
+        np.full(9, -np.inf),
+        np.full(9, np.inf),
         "result_logit_ibm",
     ),
     (
         criterion_and_derivative_probit_ibm,
         np.zeros(9),
-        -np.ones(9) * np.inf,
-        np.ones(9) * np.inf,
+        np.full(9, -np.inf),
+        np.full(9, np.inf),
         "result_probit_ibm",
     ),
 ]
@@ -357,7 +357,7 @@ def test_maximum_likelihood_bounded(
     lower_bounds,
     upper_bounds,
     expected,
-    decimals,
+    digits,
     request,
 ):
     params_expected = request.getfixturevalue(expected)
@@ -371,4 +371,4 @@ def test_maximum_likelihood_bounded(
         stopping_max_iterations=200,
     )
 
-    aaae(result_bhhh["solution_x"], params_expected.params, decimal=decimals)
+    aaae(result_bhhh["solution_x"], params_expected.params, decimal=digits)

From ae34e79e4037d624915346221c14330067181ba9 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Mon, 29 Aug 2022 21:15:22 +0200
Subject: [PATCH 06/12] Renaming.

---
 src/estimagic/optimization/bhhh.py | 170 +++++++++--------------------
 tests/optimization/test_bhhh.py    |  87 +++++++--------
 2 files changed, 90 insertions(+), 167 deletions(-)

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 5c463f481..84986a765 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -16,7 +16,7 @@ def bhhh(
     upper_bounds,
     *,
     convergence_absolute_gradient_tolerance=1e-8,
-    convergence_relative_gradient_tolerance=1e-8,
+    convergence_relative_params_tolerance=1e-8,
     stopping_max_iterations=200,
 ):
     """Minimize a likelihood function using the box-constrained BHHH algorithm.
@@ -27,10 +27,7 @@ def bhhh(
         result_dict = bhhh_unconstrained(
             criterion_and_derivative,
             x,
-            lower_bounds,
-            upper_bounds,
             convergence_absolute_gradient_tolerance,
-            convergence_relative_gradient_tolerance,
             stopping_max_iterations,
         )
     else:
@@ -39,8 +36,7 @@ def bhhh(
             x,
             lower_bounds,
             upper_bounds,
-            convergence_absolute_gradient_tolerance,
-            convergence_relative_gradient_tolerance,
+            convergence_relative_params_tolerance,
             stopping_max_iterations,
         )
 
@@ -50,13 +46,10 @@ def bhhh(
 def bhhh_unconstrained(
     criterion_and_derivative,
     x,
-    lower_bounds,
-    upper_bounds,
     convergence_absolute_gradient_tolerance,
-    convergence_relative_gradient_tolerance,
     stopping_max_iterations,
 ):
-    """Minimize a likelihood function using the bounded BHHH algorithm.
+    """Minimize a likelihood function using the unconstrained BHHH algorithm.
 
     Args:
         criterion_and_derivative (callable): A function returning the tuple:
@@ -67,12 +60,10 @@ def bhhh_unconstrained(
             for the parameter vector x.
         upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
             for the parameter vector x
-        convergence_absolute_gradient_tolerance (float): Stopping criterion for the
-            absolute gradient tolerance.
-        convergence_relative_gradient_tolerance (float): Stopping criterion for the
-            relative gradient tolerance.
-        stopping_max_iterations (int): Maximum number of iterations. If reached,
-            terminate.
+        convergence_absolute_gradient_tolerance (float): Stop if all elements of the
+            gradient are smaller than this.
+        stopping_max_iterations (int): If the maximum number of iterations is reached,
+            the optimization stops, but we do not count this as convergence.
 
     Returns:
         (dict) Result dictionary containing:
@@ -85,100 +76,52 @@ def bhhh_unconstrained(
             solution vector or reaching stopping_max_iterations.
         - message (str): Message to the user. Currently it says: "Under development."
     """
-    tol = {
-        "abs": convergence_absolute_gradient_tolerance,
-        "rel": convergence_relative_gradient_tolerance,
-    }
-    n_params = len(x)
+    criterion_accepted, jacobian = criterion_and_derivative(x)
 
-    x_accepted = x
-    criterion_accepted, jacobian = criterion_and_derivative(x_accepted)
     gradient = np.sum(jacobian, axis=0)
-
-    norm_pg0 = np.linalg.norm(
-        x_accepted - np.clip(x_accepted - gradient, lower_bounds, upper_bounds)
-    )
-    norm_pg = norm_pg0
-
-    inactive_set = estimate_epsilon_inactive_set(
-        x_accepted, norm_pg, lower_bounds, upper_bounds
-    )
-
-    gradient_reduced = gradient[inactive_set]
     hessian_approx = jacobian.T @ jacobian
-    hessian_reduced = hessian_approx[inactive_set[:, np.newaxis], inactive_set]
-
-    direction = determine_descent_direction(
-        gradient, gradient_reduced, hessian_reduced, inactive_set, n_params
-    )
+    direction = np.linalg.solve(hessian_approx, gradient)
 
     initial_step_size = 1
     step_size = initial_step_size
 
     for _n_iter in range(stopping_max_iterations):
 
-        x_candidate = np.clip(
-            x_accepted + step_size * direction, lower_bounds, upper_bounds
-        )
+        x_candidate = x + step_size * direction
 
         criterion_candidate, jacobian = criterion_and_derivative(x_candidate)
 
         if step_size == initial_step_size:
             hessian_approx = jacobian.T @ jacobian
-            hessian_reduced = hessian_approx[inactive_set[:, np.newaxis], inactive_set]
 
         if np.sum(criterion_candidate) > np.sum(criterion_accepted):
             step_size /= 2
 
             if step_size <= 0.01:
-                x_accepted = x_candidate
+                x = x_candidate
                 criterion_accepted = criterion_candidate
 
                 step_size = initial_step_size
         else:
-            x_accepted = x_candidate
+            x = x_candidate
             criterion_accepted = criterion_candidate
 
             gradient = np.sum(jacobian, axis=0)
-            gradient_reduced = gradient[inactive_set]
-
-            direction = determine_descent_direction(
-                gradient,
-                gradient_reduced,
-                hessian_reduced,
-                inactive_set,
-                n_params,
-            )
-
-            norm_pg = np.linalg.norm(
-                x_accepted - np.clip(x_accepted - gradient, lower_bounds, upper_bounds)
-            )
-            inactive_set = estimate_epsilon_inactive_set(
-                x_accepted, norm_pg, lower_bounds, upper_bounds
-            )
-
-            if norm_pg < 0:
-                gradient_reduced = gradient[inactive_set]
+            direction = np.linalg.solve(hessian_approx, gradient)
+
+            norm_grad = gradient @ direction
+
+            if norm_grad < 0:
                 hessian_approx = jacobian.T @ jacobian
-                hessian_reduced = hessian_approx[
-                    inactive_set[:, np.newaxis], inactive_set
-                ]
-
-                direction = determine_descent_direction(
-                    gradient,
-                    gradient_reduced,
-                    hessian_reduced,
-                    inactive_set,
-                    n_params,
-                )
+                direction = np.linalg.solve(hessian_approx, gradient)
 
             step_size = initial_step_size
 
-        if norm_pg < tol["abs"] + tol["rel"] * norm_pg0:
+        if norm_grad < convergence_absolute_gradient_tolerance:
             break
 
     result_dict = {
-        "solution_x": x_accepted,
+        "solution_x": x,
         "solution_criterion": criterion_accepted,
         "solution_loglikelihood": np.sum(criterion_accepted),
         "n_iterations": _n_iter,
@@ -193,7 +136,7 @@ def bhhh_box_constrained(
     x,
     lower_bounds,
     upper_bounds,
-    convergence_absolute_gradient_tolerance,
+    convergence_relative_params_tolerance,
     stopping_max_iterations,
 ):
     """Minimize a likelihood function using the box-constrained BHHH algorithm.
@@ -207,12 +150,10 @@ def bhhh_box_constrained(
             for the parameter vector x.
         upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
             for the parameter vector x
-        convergence_absolute_gradient_tolerance (float): Stopping criterion for the
-            absolute gradient tolerance.
-        convergence_relative_gradient_tolerance (float): Stopping criterion for the
-            relative gradient tolerance.
-        stopping_max_iterations (int): Maximum number of iterations. If reached,
-            terminate.
+        convergence_relative_params_tolerance (float): Stop when the relative movement
+            between parameter vectors is smaller than this.
+        stopping_max_iterations (int): If the maximum number of iterations is reached,
+            the optimization stops, but we do not count this as convergence.
 
     Returns:
         (dict) Result dictionary containing:
@@ -225,23 +166,15 @@ def bhhh_box_constrained(
             solution vector or reaching stopping_max_iterations.
         - message (str): Message to the user. Currently it says: "Under development."
     """
-    _zero_threshold = np.finfo(float).eps
     n_params = len(x)
-    step_min = 1e-6
-
-    critic_limit = convergence_absolute_gradient_tolerance
-
-    crit = 1
-    step_len_optimal = 1
+    min_step_size = 1e-6
+    _zero_threshold = np.finfo(float).eps
 
     jacobian = criterion_and_derivative(x, task="derivative")
     gradient = np.sum(jacobian, axis=0)
 
-    norm_proj_grad = np.linalg.norm(
-        x - np.clip(x - gradient, lower_bounds, upper_bounds)
-    )
     inactive_set = estimate_epsilon_inactive_set(
-        x, norm_proj_grad, lower_bounds, upper_bounds
+        x, gradient, lower_bounds, upper_bounds
     )
 
     for _n_iter in range(stopping_max_iterations):
@@ -261,7 +194,7 @@ def bhhh_box_constrained(
             direction_projected,
             lower_bounds,
             upper_bounds,
-            step_min,
+            min_step_size,
             criterion_and_derivative,
         )
 
@@ -269,21 +202,17 @@ def bhhh_box_constrained(
             x + step_len_optimal * direction_projected, lower_bounds, upper_bounds
         )
 
-        crit = np.max(
-            np.abs(((x_candidate - x) + _zero_threshold) / (x + _zero_threshold))
-            / step_len_optimal
+        relative_params_difference = np.max(
+            np.abs(((x_candidate - x)) / (x + _zero_threshold)) / step_len_optimal
         )
 
         x = x_candidate
 
-        if crit < critic_limit:
+        if relative_params_difference < convergence_relative_params_tolerance:
             break
 
-        norm_proj_grad = np.linalg.norm(
-            x - np.clip(x - gradient, lower_bounds, upper_bounds)
-        )
         inactive_set = estimate_epsilon_inactive_set(
-            x, norm_proj_grad, lower_bounds, upper_bounds
+            x, gradient, lower_bounds, upper_bounds
         )
 
     solution_criterion = criterion_and_derivative(x, task="criterion")
@@ -299,20 +228,23 @@ def bhhh_box_constrained(
     return result_dict
 
 
-def estimate_epsilon_inactive_set(x, norm_gradient, lower_bounds, upper_bounds):
+def estimate_epsilon_inactive_set(x, gradient, lower_bounds, upper_bounds):
     """Estimate the set of epsilon-inactive bound constraints up to a tolerance.
 
     The set of epsilon-inactive indices underestimates (overestimates) the actual
     set of inactive (active) indices.
 
-        x (np.ndarray): Current candidate vector.
-        norm_gradient (float): Norm of the projected gradient.
+        x (np.ndarray): Current parameter vector of shape (n_params,).
+        gradient (np.ndarray): Current gradient vector of shape (n_params,).
         lower_bounds (np.ndarray): 1d array of shape (n_params,) with lower bounds
             for the parameter vector x.
         upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
             for the parameter vector x
     """
-    epsilon = min(np.min(upper_bounds - lower_bounds) / 2, norm_gradient)
+    norm_proj_grad = np.linalg.norm(
+        x - np.clip(x - gradient, lower_bounds, upper_bounds)
+    )
+    epsilon = min(np.min(upper_bounds - lower_bounds) / 2, norm_proj_grad)
 
     inactive_set = np.where(
         (lower_bounds + epsilon < x) & (x < upper_bounds - epsilon)
@@ -343,34 +275,36 @@ def find_optimal_step_len(
     direction_projected,
     lower_bounds,
     upper_bounds,
-    step_min,
+    min_step_size,
     criterion_and_derivative,
 ):
     """Find optimal step length."""
     step_len_trial = 2
     step_len_optimal = 1
 
-    loglike_higher = 1
-    loglike_lower = 0
+    loglike_full_step = 1
+    loglike_half_step = 0
 
-    while (loglike_higher > loglike_lower) & (step_len_trial >= step_min):
-        step_len_trial = step_len_trial / 2
+    while (loglike_full_step > loglike_half_step) & (step_len_trial >= min_step_size):
+        step_len_trial /= 2
 
-        criterion_candidate_higher = criterion_and_derivative(
+        criterion_full_step = criterion_and_derivative(
             np.clip(
                 x + step_len_trial * direction_projected, lower_bounds, upper_bounds
             ),
             task="criterion",
         )
-        criterion_candidate_lower = criterion_and_derivative(
+        criterion_half_step = criterion_and_derivative(
             np.clip(
-                x + step_len_trial * direction_projected / 2, lower_bounds, upper_bounds
+                x + (step_len_trial / 2) * direction_projected,
+                lower_bounds,
+                upper_bounds,
             ),
             task="criterion",
         )
 
-        loglike_higher = np.sum(criterion_candidate_higher)
-        loglike_lower = np.sum(criterion_candidate_lower)
+        loglike_full_step = np.sum(criterion_full_step)
+        loglike_half_step = np.sum(criterion_half_step)
 
         step_len_optimal = step_len_trial
 
diff --git a/tests/optimization/test_bhhh.py b/tests/optimization/test_bhhh.py
index 6260daa58..36a65a3a5 100644
--- a/tests/optimization/test_bhhh.py
+++ b/tests/optimization/test_bhhh.py
@@ -30,7 +30,7 @@ def generate_test_data():
     return endog, exog
 
 
-def ibm_test_data():
+def load_ibm_data():
     df = pd.read_csv(TEST_FIXTURES_DIR / "telco_churn_clean.csv")
 
     exog = df.drop(columns="Churn").to_numpy()
@@ -127,7 +127,7 @@ def criterion_and_derivative_logit_ibm(x, task="criterion_and_derivative"):
             which is 2d numpy array.
             If task=="criterion_and_derivative" it returns both as a tuple.
     """
-    endog, exog = ibm_test_data()
+    endog, exog = load_ibm_data()
     result = ()
 
     if "criterion" in task:
@@ -191,7 +191,7 @@ def criterion_and_derivative_probit_ibm(x, task="criterion_and_derivative"):
             which is 2d numpy array.
             If task=="criterion_and_derivative" it returns both as a tuple.
     """
-    endog, exog = ibm_test_data()
+    endog, exog = load_ibm_data()
     result = ()
 
     if "criterion" in task:
@@ -213,102 +213,94 @@ def criterion_and_derivative_probit_ibm(x, task="criterion_and_derivative"):
 
 
 @pytest.fixture
-def result_logit_unbounded():
+def result_logit_unconstrained():
     endog, exog = generate_test_data()
-    result_unbounded = sm.Logit(endog, exog).fit(disp=True)
+    result = sm.Logit(endog, exog).fit(disp=True)
 
-    return result_unbounded
+    return result.params
 
 
 @pytest.fixture
-def result_probit_unbounded():
+def result_probit_unconstrained():
     endog, exog = generate_test_data()
-    result_unbounded = sm.Probit(endog, exog).fit(disp=True)
+    result = sm.Probit(endog, exog).fit(disp=True)
 
-    return result_unbounded
+    return result.params
 
 
 @pytest.fixture
-def result_logit_bounded():
+def result_logit_constrained():
     endog, exog = generate_test_data()
-    result_bounded = sm.Logit(endog, exog).fit(
-        method="lbfgs", bounds=((-5, np.inf), (-10, 10), (-10, 10)), disp=False
+    result = sm.Logit(endog, exog).fit(
+        method="lbfgs", bounds=((-5, 10), (-100, 100), (-100, 100)), disp=False
     )
 
-    return result_bounded
+    return result.params
 
 
 @pytest.fixture
-def result_probit_bounded():
+def result_probit_constrained():
     endog, exog = generate_test_data()
-    result_bounded = sm.Logit(endog, exog).fit(
-        method="lbfgs", bounds=((-5, np.inf), (-10, 10), (-10, 10)), disp=False
+    result = sm.Logit(endog, exog).fit(
+        method="lbfgs",
+        bounds=((-5, np.inf), (-np.inf, np.inf), (-np.inf, np.inf)),
+        disp=False,
     )
 
-    return result_bounded
+    return result.params
 
 
 @pytest.fixture
 def result_logit_ibm():
-    endog, exog = ibm_test_data()
-    result_unbounded = sm.Logit(endog, exog).fit(disp=False)
+    endog, exog = load_ibm_data()
+    result = sm.Logit(endog, exog).fit(disp=False)
 
-    return result_unbounded
+    return result.params
 
 
 @pytest.fixture
 def result_probit_ibm():
-    endog, exog = ibm_test_data()
-    result_unbounded = sm.Probit(endog, exog).fit(disp=False)
+    endog, exog = load_ibm_data()
+    result = sm.Probit(endog, exog).fit(disp=False)
 
-    return result_unbounded
+    return result.params
 
 
 # =====================================================================================
 # Tests
 # =====================================================================================
 
-TEST_CASES_UNBOUNDED = [
+TEST_CASES_UNCONSTRAINED = [
     (
         criterion_and_derivative_logit,
         np.zeros(3),
-        np.full(3, -np.inf),
-        np.full(3, np.inf),
-        "result_logit_unbounded",
+        "result_logit_unconstrained",
     ),
     (
         criterion_and_derivative_probit,
         np.zeros(3),
-        np.full(3, -np.inf),
-        np.full(3, np.inf),
-        "result_probit_unbounded",
+        "result_probit_unconstrained",
     ),
     (
         criterion_and_derivative_logit_ibm,
         np.zeros(9),
-        np.full(9, -np.inf),
-        np.full(9, np.inf),
         "result_logit_ibm",
     ),
     (
         criterion_and_derivative_probit_ibm,
         np.zeros(9),
-        np.full(9, -np.inf),
-        np.full(9, np.inf),
         "result_probit_ibm",
     ),
 ]
 
 
 @pytest.mark.parametrize(
-    "criterion_and_derivative, x0, lower_bounds, upper_bounds, expected",
-    TEST_CASES_UNBOUNDED,
+    "criterion_and_derivative, x0, expected",
+    TEST_CASES_UNCONSTRAINED,
 )
 def test_maximum_likelihood(
     criterion_and_derivative,
     x0,
-    lower_bounds,
-    upper_bounds,
     expected,
     request,
 ):
@@ -317,23 +309,20 @@ def test_maximum_likelihood(
     result_bhhh = bhhh_unconstrained(
         criterion_and_derivative,
         x=x0,
-        lower_bounds=lower_bounds,
-        upper_bounds=upper_bounds,
         convergence_absolute_gradient_tolerance=1e-8,
-        convergence_relative_gradient_tolerance=1e-8,
         stopping_max_iterations=200,
     )
 
-    aaae(result_bhhh["solution_x"], params_expected.params, decimal=4)
+    aaae(result_bhhh["solution_x"], params_expected, decimal=4)
 
 
-TEST_CASES_BOUNDED = [
+TEST_CASES_CONSTRAINED = [
     (
         criterion_and_derivative_logit,
         np.zeros(3),
         np.array([-5, -100, -100]),
-        np.array([100, 100, 100]),
-        "result_logit_bounded",
+        np.array([10, 100, 100]),
+        "result_logit_constrained",
         4,
     ),
     (
@@ -341,7 +330,7 @@ def test_maximum_likelihood(
         np.zeros(3),
         np.array([-5, -np.inf, -np.inf]),
         np.array([10, np.inf, np.inf]),
-        "result_probit_bounded",
+        "result_probit_constrained",
         0,
     ),
 ]
@@ -349,7 +338,7 @@ def test_maximum_likelihood(
 
 @pytest.mark.parametrize(
     "criterion_and_derivative, x0, lower_bounds, upper_bounds, expected, digits",
-    TEST_CASES_BOUNDED,
+    TEST_CASES_CONSTRAINED,
 )
 def test_maximum_likelihood_bounded(
     criterion_and_derivative,
@@ -367,8 +356,8 @@ def test_maximum_likelihood_bounded(
         x=x0,
         lower_bounds=lower_bounds,
         upper_bounds=upper_bounds,
-        convergence_absolute_gradient_tolerance=1e-4,
+        convergence_relative_params_tolerance=1e-4,
         stopping_max_iterations=200,
     )
 
-    aaae(result_bhhh["solution_x"], params_expected.params, decimal=digits)
+    aaae(result_bhhh["solution_x"], params_expected, decimal=digits)

From d11d614d6160b5de40d1b4c37d2195de14c24e11 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Tue, 30 Aug 2022 15:47:14 +0200
Subject: [PATCH 07/12] Harmonize stopping criteria.

---
 src/estimagic/optimization/bhhh.py | 56 ++++++++++++++++++++++--------
 tests/optimization/test_bhhh.py    | 20 ++++++++---
 2 files changed, 58 insertions(+), 18 deletions(-)

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 84986a765..245c31d83 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -27,6 +27,7 @@ def bhhh(
         result_dict = bhhh_unconstrained(
             criterion_and_derivative,
             x,
+            convergence_relative_params_tolerance,
             convergence_absolute_gradient_tolerance,
             stopping_max_iterations,
         )
@@ -37,6 +38,7 @@ def bhhh(
             lower_bounds,
             upper_bounds,
             convergence_relative_params_tolerance,
+            convergence_absolute_gradient_tolerance,
             stopping_max_iterations,
         )
 
@@ -46,6 +48,7 @@ def bhhh(
 def bhhh_unconstrained(
     criterion_and_derivative,
     x,
+    convergence_relative_params_tolerance,
     convergence_absolute_gradient_tolerance,
     stopping_max_iterations,
 ):
@@ -60,6 +63,8 @@ def bhhh_unconstrained(
             for the parameter vector x.
         upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
             for the parameter vector x
+        convergence_relative_params_tolerance (float): Stop when the relative movement
+            between parameter vectors is smaller than this.
         convergence_absolute_gradient_tolerance (float): Stop if all elements of the
             gradient are smaller than this.
         stopping_max_iterations (int): If the maximum number of iterations is reached,
@@ -76,18 +81,22 @@ def bhhh_unconstrained(
             solution vector or reaching stopping_max_iterations.
         - message (str): Message to the user. Currently it says: "Under development."
     """
+    _zero_threshold = np.finfo(float).eps
+
     criterion_accepted, jacobian = criterion_and_derivative(x)
 
     gradient = np.sum(jacobian, axis=0)
     hessian_approx = jacobian.T @ jacobian
     direction = np.linalg.solve(hessian_approx, gradient)
 
+    norm_grad = gradient @ direction
+
     initial_step_size = 1
     step_size = initial_step_size
 
-    for _n_iter in range(stopping_max_iterations):
+    x_candidate = x + step_size * direction
 
-        x_candidate = x + step_size * direction
+    for _n_iter in range(stopping_max_iterations):
 
         criterion_candidate, jacobian = criterion_and_derivative(x_candidate)
 
@@ -117,13 +126,23 @@ def bhhh_unconstrained(
 
             step_size = initial_step_size
 
-        if norm_grad < convergence_absolute_gradient_tolerance:
+        x_candidate = x + step_size * direction
+
+        relative_params_difference = np.max(
+            np.abs(((x_candidate - x)) / (x + _zero_threshold)) / step_size
+        )
+
+        if relative_params_difference < convergence_relative_params_tolerance:
+            break
+        elif norm_grad < convergence_absolute_gradient_tolerance:
             break
 
     result_dict = {
         "solution_x": x,
         "solution_criterion": criterion_accepted,
         "solution_loglikelihood": np.sum(criterion_accepted),
+        "relative_params_difference": relative_params_difference,
+        "absolute_gradient_norm": norm_grad,
         "n_iterations": _n_iter,
         "message": "Under develpment",
     }
@@ -137,6 +156,7 @@ def bhhh_box_constrained(
     lower_bounds,
     upper_bounds,
     convergence_relative_params_tolerance,
+    convergence_absolute_gradient_tolerance,
     stopping_max_iterations,
 ):
     """Minimize a likelihood function using the box-constrained BHHH algorithm.
@@ -152,6 +172,8 @@ def bhhh_box_constrained(
             for the parameter vector x
         convergence_relative_params_tolerance (float): Stop when the relative movement
             between parameter vectors is smaller than this.
+        convergence_absolute_gradient_tolerance (float): Stop if all elements of the
+            projected gradient are smaller than this.
         stopping_max_iterations (int): If the maximum number of iterations is reached,
             the optimization stops, but we do not count this as convergence.
 
@@ -173,8 +195,11 @@ def bhhh_box_constrained(
     jacobian = criterion_and_derivative(x, task="derivative")
     gradient = np.sum(jacobian, axis=0)
 
+    norm_proj_grad = np.linalg.norm(
+        x - np.clip(x - gradient, lower_bounds, upper_bounds)
+    )
     inactive_set = estimate_epsilon_inactive_set(
-        x, gradient, lower_bounds, upper_bounds
+        x, norm_proj_grad, lower_bounds, upper_bounds
     )
 
     for _n_iter in range(stopping_max_iterations):
@@ -201,18 +226,22 @@ def bhhh_box_constrained(
         x_candidate = np.clip(
             x + step_len_optimal * direction_projected, lower_bounds, upper_bounds
         )
-
         relative_params_difference = np.max(
             np.abs(((x_candidate - x)) / (x + _zero_threshold)) / step_len_optimal
         )
-
         x = x_candidate
 
+        norm_proj_grad = np.linalg.norm(
+            x - np.clip(x - gradient, lower_bounds, upper_bounds)
+        )
+
         if relative_params_difference < convergence_relative_params_tolerance:
             break
+        elif norm_proj_grad < convergence_absolute_gradient_tolerance:
+            break
 
         inactive_set = estimate_epsilon_inactive_set(
-            x, gradient, lower_bounds, upper_bounds
+            x, norm_proj_grad, lower_bounds, upper_bounds
         )
 
     solution_criterion = criterion_and_derivative(x, task="criterion")
@@ -221,6 +250,8 @@ def bhhh_box_constrained(
         "solution_x": x,
         "solution_criterion": solution_criterion,
         "solution_loglikelihood": np.sum(solution_criterion),
+        "relative_params_difference": relative_params_difference,
+        "absolute_gradient_norm": norm_proj_grad,
         "n_iterations": _n_iter,
         "message": "Under develpment",
     }
@@ -228,23 +259,20 @@ def bhhh_box_constrained(
     return result_dict
 
 
-def estimate_epsilon_inactive_set(x, gradient, lower_bounds, upper_bounds):
-    """Estimate the set of epsilon-inactive bound constraints up to a tolerance.
+def estimate_epsilon_inactive_set(x, norm_grad, lower_bounds, upper_bounds):
+    """Estimate the set of epsilon-inactive bound constraints.
 
     The set of epsilon-inactive indices underestimates (overestimates) the actual
     set of inactive (active) indices.
 
         x (np.ndarray): Current parameter vector of shape (n_params,).
-        gradient (np.ndarray): Current gradient vector of shape (n_params,).
+        norm_grad (float): Norm of the projected gradient.
         lower_bounds (np.ndarray): 1d array of shape (n_params,) with lower bounds
             for the parameter vector x.
         upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
             for the parameter vector x
     """
-    norm_proj_grad = np.linalg.norm(
-        x - np.clip(x - gradient, lower_bounds, upper_bounds)
-    )
-    epsilon = min(np.min(upper_bounds - lower_bounds) / 2, norm_proj_grad)
+    epsilon = min(np.min(upper_bounds - lower_bounds) / 2, norm_grad)
 
     inactive_set = np.where(
         (lower_bounds + epsilon < x) & (x < upper_bounds - epsilon)
diff --git a/tests/optimization/test_bhhh.py b/tests/optimization/test_bhhh.py
index 36a65a3a5..756456849 100644
--- a/tests/optimization/test_bhhh.py
+++ b/tests/optimization/test_bhhh.py
@@ -306,14 +306,25 @@ def test_maximum_likelihood(
 ):
     params_expected = request.getfixturevalue(expected)
 
-    result_bhhh = bhhh_unconstrained(
+    result_unconstr = bhhh_unconstrained(
         criterion_and_derivative,
         x=x0,
+        convergence_relative_params_tolerance=1e-6,
+        convergence_absolute_gradient_tolerance=1e-8,
+        stopping_max_iterations=200,
+    )
+    result_constr = bhhh_box_constrained(
+        criterion_and_derivative,
+        x=x0,
+        lower_bounds=np.full(len(x0), -np.inf),
+        upper_bounds=np.full(len(x0), np.inf),
+        convergence_relative_params_tolerance=1e-6,
         convergence_absolute_gradient_tolerance=1e-8,
         stopping_max_iterations=200,
     )
 
-    aaae(result_bhhh["solution_x"], params_expected, decimal=4)
+    aaae(result_unconstr["solution_x"], params_expected, decimal=4)
+    aaae(result_constr["solution_x"], params_expected, decimal=4)
 
 
 TEST_CASES_CONSTRAINED = [
@@ -351,13 +362,14 @@ def test_maximum_likelihood_bounded(
 ):
     params_expected = request.getfixturevalue(expected)
 
-    result_bhhh = bhhh_box_constrained(
+    result = bhhh_box_constrained(
         criterion_and_derivative,
         x=x0,
         lower_bounds=lower_bounds,
         upper_bounds=upper_bounds,
         convergence_relative_params_tolerance=1e-4,
+        convergence_absolute_gradient_tolerance=1e-4,
         stopping_max_iterations=200,
     )
 
-    aaae(result_bhhh["solution_x"], params_expected, decimal=digits)
+    aaae(result["solution_x"], params_expected, decimal=digits)

From 61163efb0b3bcf2f4c18cc9a46dc2b2d4e512a64 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Tue, 30 Aug 2022 18:32:38 +0200
Subject: [PATCH 08/12] Update documentation.

---
 docs/source/algorithms.md          | 23 ++++++++++++++++-------
 docs/source/refs.bib               |  9 +++++++++
 src/estimagic/optimization/bhhh.py | 28 ++++++++++++++++------------
 3 files changed, 41 insertions(+), 19 deletions(-)

diff --git a/docs/source/algorithms.md b/docs/source/algorithms.md
index 871e9d70c..7eeb18c10 100644
--- a/docs/source/algorithms.md
+++ b/docs/source/algorithms.md
@@ -578,14 +578,20 @@ We implement a few algorithms from scratch. They are currently considered experi
 
         "bhhh"
 
-    Minimize a likelihood function using the bounded BHHH algorithm.
+    Minimize a likelihood function using the box-constraint BHHH algorithm.
 
     BHHH (:cite:`Berndt1974`) can - and should ONLY - be used for minimizing
-    (or maximizing) a likelihood. It is similar to the Newton-Raphson
+    (or maximizing) a likelihood function. It is similar to the Newton-Raphson
     algorithm, but replaces the Hessian matrix with the outer product of the
     gradient. This approximation is based on the information matrix equality
     (:cite:`Halbert1982`) and is thus only vaid when minimizing (or maximizing)
-    a likelihood. Bounds, i.e. box constraints, are supported.
+    a likelihood.
+
+    Bounds, i.e. box constraints, are supported. In order to identify the active
+    constraints in the set of inequality constraints, an epsilon-active-set approach is
+    used (see, e.g. :cite:`Nocedal2006`, p. 308, for the active-set method in general and
+    :cite:`Kelley1999`, p. 97, on the estimation of epsilon-active sets a la the
+    Projected BFGS–Armijo algorithm).
 
     The criterion function :func:`func` should return a dictionary with
     at least the entry ``{"contributions": array_or_pytree}`` where ``array_or_pytree``
@@ -593,10 +599,13 @@ We implement a few algorithms from scratch. They are currently considered experi
 
     bhhh supports the following options:
 
-    - **convergence_absolute_gradient_tolerance** (float): Stopping criterion for the
-      gradient tolerance. Default is 1e-8.
-    - **stopping_max_iterations** (int): Maximum number of iterations.
-      If reached, terminate. Default is 200.
+    - **convergence.relative_params_tolerance** (float): Stop when the relative movement
+            between parameter vectors is smaller than this. The default is 1e-8.
+    - **convergence.absolute_gradient_tolerance** (float): Stop if all elements of the
+            projected gradient are smaller than this. The default is 1e-8.
+    - **stopping.max_iterations** (int): If the maximum number of iterations is reached,
+      the optimization stops, but we do not count this as convergence.
+      The default is 200.
 
 ```
 
diff --git a/docs/source/refs.bib b/docs/source/refs.bib
index 0ee68d3d3..bbc713ac0 100644
--- a/docs/source/refs.bib
+++ b/docs/source/refs.bib
@@ -884,4 +884,13 @@ @article{Zhang2010
 URL = {https://doi.org/10.1137/09075531X},
 }
 
+@book{Kelley1999,
+author = {Kelley, C. T.},
+title = {Iterative Methods for Optimization},
+publisher = {Society for Industrial and Applied Mathematics},
+year = {1999},
+doi = {10.1137/1.9781611970920},
+URL = {https://epubs.siam.org/doi/abs/10.1137/1.9781611970920},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}
diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 245c31d83..2345be6c7 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -161,6 +161,10 @@ def bhhh_box_constrained(
 ):
     """Minimize a likelihood function using the box-constrained BHHH algorithm.
 
+    The (in)active constraints are identified via an epsilon-active-set method,
+    similar to the approach used by the Projected BFGS-Armijo algorithm
+    (see :cite:`Kelley1999`, p. 97).
+
     Args:
         criterion_and_derivative (callable): A function returning the tuple:
             - criterion (np.ndarray): Likelihood contributions of shape (n_obs,)
@@ -214,7 +218,7 @@ def bhhh_box_constrained(
             gradient, gradient_reduced, hessian_reduced, inactive_set, n_params
         )
 
-        step_len_optimal = find_optimal_step_len(
+        step_size = find_optimal_step_size(
             x,
             direction_projected,
             lower_bounds,
@@ -224,10 +228,10 @@ def bhhh_box_constrained(
         )
 
         x_candidate = np.clip(
-            x + step_len_optimal * direction_projected, lower_bounds, upper_bounds
+            x + step_size * direction_projected, lower_bounds, upper_bounds
         )
         relative_params_difference = np.max(
-            np.abs(((x_candidate - x)) / (x + _zero_threshold)) / step_len_optimal
+            np.abs(((x_candidate - x)) / (x + _zero_threshold)) / step_size
         )
         x = x_candidate
 
@@ -298,7 +302,7 @@ def determine_descent_direction(
     return direction_all
 
 
-def find_optimal_step_len(
+def find_optimal_step_size(
     x,
     direction_projected,
     lower_bounds,
@@ -307,24 +311,24 @@ def find_optimal_step_len(
     criterion_and_derivative,
 ):
     """Find optimal step length."""
-    step_len_trial = 2
-    step_len_optimal = 1
+    step_size_trial = 2
+    step_size_optimal = 1
 
     loglike_full_step = 1
     loglike_half_step = 0
 
-    while (loglike_full_step > loglike_half_step) & (step_len_trial >= min_step_size):
-        step_len_trial /= 2
+    while (loglike_full_step > loglike_half_step) & (step_size_trial >= min_step_size):
+        step_size_trial /= 2
 
         criterion_full_step = criterion_and_derivative(
             np.clip(
-                x + step_len_trial * direction_projected, lower_bounds, upper_bounds
+                x + step_size_trial * direction_projected, lower_bounds, upper_bounds
             ),
             task="criterion",
         )
         criterion_half_step = criterion_and_derivative(
             np.clip(
-                x + (step_len_trial / 2) * direction_projected,
+                x + (step_size_trial / 2) * direction_projected,
                 lower_bounds,
                 upper_bounds,
             ),
@@ -334,6 +338,6 @@ def find_optimal_step_len(
         loglike_full_step = np.sum(criterion_full_step)
         loglike_half_step = np.sum(criterion_half_step)
 
-        step_len_optimal = step_len_trial
+        step_size_optimal = step_size_trial
 
-    return step_len_optimal
+    return step_size_optimal

From 730ed940ae44f1b7cfa35af39ed40e77f6fb96c5 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Wed, 31 Aug 2022 13:48:44 +0200
Subject: [PATCH 09/12] Add variable on convergence status.

---
 src/estimagic/optimization/bhhh.py | 8 ++++++++
 1 file changed, 8 insertions(+)

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 2345be6c7..80dee10bb 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -96,6 +96,7 @@ def bhhh_unconstrained(
 
     x_candidate = x + step_size * direction
 
+    converged = False
     for _n_iter in range(stopping_max_iterations):
 
         criterion_candidate, jacobian = criterion_and_derivative(x_candidate)
@@ -133,14 +134,17 @@ def bhhh_unconstrained(
         )
 
         if relative_params_difference < convergence_relative_params_tolerance:
+            converged = True
             break
         elif norm_grad < convergence_absolute_gradient_tolerance:
+            converged = True
             break
 
     result_dict = {
         "solution_x": x,
         "solution_criterion": criterion_accepted,
         "solution_loglikelihood": np.sum(criterion_accepted),
+        "converged": converged,
         "relative_params_difference": relative_params_difference,
         "absolute_gradient_norm": norm_grad,
         "n_iterations": _n_iter,
@@ -206,6 +210,7 @@ def bhhh_box_constrained(
         x, norm_proj_grad, lower_bounds, upper_bounds
     )
 
+    converged = False
     for _n_iter in range(stopping_max_iterations):
         jacobian = criterion_and_derivative(x, task="derivative")
         gradient = np.sum(jacobian, axis=0)
@@ -240,8 +245,10 @@ def bhhh_box_constrained(
         )
 
         if relative_params_difference < convergence_relative_params_tolerance:
+            converged = True
             break
         elif norm_proj_grad < convergence_absolute_gradient_tolerance:
+            converged = True
             break
 
         inactive_set = estimate_epsilon_inactive_set(
@@ -254,6 +261,7 @@ def bhhh_box_constrained(
         "solution_x": x,
         "solution_criterion": solution_criterion,
         "solution_loglikelihood": np.sum(solution_criterion),
+        "converged": converged,
         "relative_params_difference": relative_params_difference,
         "absolute_gradient_norm": norm_proj_grad,
         "n_iterations": _n_iter,

From 3f6bc889d54e4dbb4e6437e028dac750e5440546 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Wed, 31 Aug 2022 14:12:42 +0200
Subject: [PATCH 10/12] Add more tests.

---
 src/estimagic/optimization/bhhh.py | 14 ++++----
 tests/optimization/test_bhhh.py    | 55 ++++++++++++------------------
 2 files changed, 29 insertions(+), 40 deletions(-)

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 80dee10bb..8bfce4534 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -206,7 +206,7 @@ def bhhh_box_constrained(
     norm_proj_grad = np.linalg.norm(
         x - np.clip(x - gradient, lower_bounds, upper_bounds)
     )
-    inactive_set = estimate_epsilon_inactive_set(
+    inactive_set = _estimate_epsilon_inactive_set(
         x, norm_proj_grad, lower_bounds, upper_bounds
     )
 
@@ -219,11 +219,11 @@ def bhhh_box_constrained(
         hessian_approx = jacobian.T @ jacobian
         hessian_reduced = hessian_approx[inactive_set[:, np.newaxis], inactive_set]
 
-        direction_projected = determine_descent_direction(
+        direction_projected = _determine_descent_direction(
             gradient, gradient_reduced, hessian_reduced, inactive_set, n_params
         )
 
-        step_size = find_optimal_step_size(
+        step_size = _find_optimal_step_size(
             x,
             direction_projected,
             lower_bounds,
@@ -251,7 +251,7 @@ def bhhh_box_constrained(
             converged = True
             break
 
-        inactive_set = estimate_epsilon_inactive_set(
+        inactive_set = _estimate_epsilon_inactive_set(
             x, norm_proj_grad, lower_bounds, upper_bounds
         )
 
@@ -271,7 +271,7 @@ def bhhh_box_constrained(
     return result_dict
 
 
-def estimate_epsilon_inactive_set(x, norm_grad, lower_bounds, upper_bounds):
+def _estimate_epsilon_inactive_set(x, norm_grad, lower_bounds, upper_bounds):
     """Estimate the set of epsilon-inactive bound constraints.
 
     The set of epsilon-inactive indices underestimates (overestimates) the actual
@@ -293,7 +293,7 @@ def estimate_epsilon_inactive_set(x, norm_grad, lower_bounds, upper_bounds):
     return inactive_set
 
 
-def determine_descent_direction(
+def _determine_descent_direction(
     gradient_candidate, gradient_reduced, hessian_reduced, inactive_set, n_params
 ):
     """Determine the new descent (search) direction."""
@@ -310,7 +310,7 @@ def determine_descent_direction(
     return direction_all
 
 
-def find_optimal_step_size(
+def _find_optimal_step_size(
     x,
     direction_projected,
     lower_bounds,
diff --git a/tests/optimization/test_bhhh.py b/tests/optimization/test_bhhh.py
index 756456849..263ab66dc 100644
--- a/tests/optimization/test_bhhh.py
+++ b/tests/optimization/test_bhhh.py
@@ -269,57 +269,46 @@ def result_probit_ibm():
 # =====================================================================================
 # Tests
 # =====================================================================================
-
 TEST_CASES_UNCONSTRAINED = [
-    (
-        criterion_and_derivative_logit,
-        np.zeros(3),
-        "result_logit_unconstrained",
-    ),
-    (
-        criterion_and_derivative_probit,
-        np.zeros(3),
-        "result_probit_unconstrained",
-    ),
-    (
-        criterion_and_derivative_logit_ibm,
-        np.zeros(9),
-        "result_logit_ibm",
-    ),
-    (
-        criterion_and_derivative_probit_ibm,
-        np.zeros(9),
-        "result_probit_ibm",
-    ),
+    (criterion_and_derivative_logit, 3, "result_logit_unconstrained", 1e-6, 0),
+    (criterion_and_derivative_logit, 3, "result_logit_unconstrained", 0, 1e-8),
+    (criterion_and_derivative_probit, 3, "result_probit_unconstrained", 1e-6, 0),
+    (criterion_and_derivative_probit, 3, "result_probit_unconstrained", 0, 1e-8),
+    (criterion_and_derivative_logit_ibm, 9, "result_logit_ibm", 1e-6, 0),
+    (criterion_and_derivative_logit_ibm, 9, "result_logit_ibm", 0, 1e-8),
+    (criterion_and_derivative_probit_ibm, 9, "result_probit_ibm", 1e-6, 0),
+    (criterion_and_derivative_probit_ibm, 9, "result_probit_ibm", 0, 1e-8),
 ]
 
 
 @pytest.mark.parametrize(
-    "criterion_and_derivative, x0, expected",
+    "criterion_and_derivative, n_params, expected, params_tol, gradient_tol",
     TEST_CASES_UNCONSTRAINED,
 )
-def test_maximum_likelihood(
+def test_maximum_likelihood_unconstrained(
     criterion_and_derivative,
-    x0,
+    n_params,
     expected,
+    params_tol,
+    gradient_tol,
     request,
 ):
     params_expected = request.getfixturevalue(expected)
 
     result_unconstr = bhhh_unconstrained(
         criterion_and_derivative,
-        x=x0,
-        convergence_relative_params_tolerance=1e-6,
-        convergence_absolute_gradient_tolerance=1e-8,
+        x=np.zeros(n_params),
+        convergence_relative_params_tolerance=params_tol,
+        convergence_absolute_gradient_tolerance=gradient_tol,
         stopping_max_iterations=200,
     )
     result_constr = bhhh_box_constrained(
         criterion_and_derivative,
-        x=x0,
-        lower_bounds=np.full(len(x0), -np.inf),
-        upper_bounds=np.full(len(x0), np.inf),
-        convergence_relative_params_tolerance=1e-6,
-        convergence_absolute_gradient_tolerance=1e-8,
+        x=np.zeros(n_params),
+        lower_bounds=np.full(n_params, -np.inf),
+        upper_bounds=np.full(n_params, np.inf),
+        convergence_relative_params_tolerance=params_tol,
+        convergence_absolute_gradient_tolerance=gradient_tol,
         stopping_max_iterations=200,
     )
 
@@ -351,7 +340,7 @@ def test_maximum_likelihood(
     "criterion_and_derivative, x0, lower_bounds, upper_bounds, expected, digits",
     TEST_CASES_CONSTRAINED,
 )
-def test_maximum_likelihood_bounded(
+def test_maximum_likelihood_constrained(
     criterion_and_derivative,
     x0,
     lower_bounds,

From a3c2d9011b2265e1a2b7a18782aa10ffb7e6c5a3 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Fri, 21 Oct 2022 13:03:38 +0200
Subject: [PATCH 11/12] Remove constrained probit test and put references in
 docstrings.

---
 src/estimagic/optimization/bhhh.py |  9 +++++++++
 tests/optimization/test_bhhh.py    | 14 ++------------
 2 files changed, 11 insertions(+), 12 deletions(-)

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 8bfce4534..3317f39bd 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -54,6 +54,10 @@ def bhhh_unconstrained(
 ):
     """Minimize a likelihood function using the unconstrained BHHH algorithm.
 
+    This implementation is based on the Matlab code by Fedor Iskhakov, which can
+    be found on his `Github page <https://github.com/fediskhakov/MPECvsNFXP/blob/
+    8212346a0426f54977823bd754947f528ed749ba/nfxp_Rust87/nfxp.m#L315>`.
+
     Args:
         criterion_and_derivative (callable): A function returning the tuple:
             - criterion (np.ndarray): Likelihood contributions of shape (n_obs,)
@@ -169,6 +173,11 @@ def bhhh_box_constrained(
     similar to the approach used by the Projected BFGS-Armijo algorithm
     (see :cite:`Kelley1999`, p. 97).
 
+    The unconstrained version of this algorithm is based on the Matlab implementation
+    by Adam E. Theising. See `Assignment 4 <https://adamtheising.github.io/aae637>`
+    on Maximum Likelihood Estimation on the course page of Applied Econometrics II.
+
+
     Args:
         criterion_and_derivative (callable): A function returning the tuple:
             - criterion (np.ndarray): Likelihood contributions of shape (n_obs,)
diff --git a/tests/optimization/test_bhhh.py b/tests/optimization/test_bhhh.py
index 263ab66dc..9248758dd 100644
--- a/tests/optimization/test_bhhh.py
+++ b/tests/optimization/test_bhhh.py
@@ -323,21 +323,12 @@ def test_maximum_likelihood_unconstrained(
         np.array([-5, -100, -100]),
         np.array([10, 100, 100]),
         "result_logit_constrained",
-        4,
-    ),
-    (
-        criterion_and_derivative_probit,
-        np.zeros(3),
-        np.array([-5, -np.inf, -np.inf]),
-        np.array([10, np.inf, np.inf]),
-        "result_probit_constrained",
-        0,
     ),
 ]
 
 
 @pytest.mark.parametrize(
-    "criterion_and_derivative, x0, lower_bounds, upper_bounds, expected, digits",
+    "criterion_and_derivative, x0, lower_bounds, upper_bounds, expected",
     TEST_CASES_CONSTRAINED,
 )
 def test_maximum_likelihood_constrained(
@@ -346,7 +337,6 @@ def test_maximum_likelihood_constrained(
     lower_bounds,
     upper_bounds,
     expected,
-    digits,
     request,
 ):
     params_expected = request.getfixturevalue(expected)
@@ -361,4 +351,4 @@ def test_maximum_likelihood_constrained(
         stopping_max_iterations=200,
     )
 
-    aaae(result["solution_x"], params_expected, decimal=digits)
+    aaae(result["solution_x"], params_expected, decimal=4)

From c78c111030e967e27fa9b7c7dcdf378898c17c78 Mon Sep 17 00:00:00 2001
From: Sebastian Gsell <sebastian.gsell93@gmail.com>
Date: Fri, 21 Oct 2022 22:02:34 +0200
Subject: [PATCH 12/12] Remove constrained probit test.

---
 src/estimagic/optimization/bhhh.py            | 11 +++--
 tests/optimization/test_bhhh.py               | 40 +++++++------------
 .../optimization/test_pounders_integration.py | 17 --------
 3 files changed, 22 insertions(+), 46 deletions(-)

diff --git a/src/estimagic/optimization/bhhh.py b/src/estimagic/optimization/bhhh.py
index 3317f39bd..f3c0a7fcc 100644
--- a/src/estimagic/optimization/bhhh.py
+++ b/src/estimagic/optimization/bhhh.py
@@ -75,7 +75,7 @@ def bhhh_unconstrained(
             the optimization stops, but we do not count this as convergence.
 
     Returns:
-        (dict) Result dictionary containing:
+        dict: Result dictionary containing:
 
         - solution_x (np.ndarray): Solution vector of shape (n_params,).
         - solution_criterion (np.ndarray): Likelihood contributions at the solution.
@@ -195,7 +195,7 @@ def bhhh_box_constrained(
             the optimization stops, but we do not count this as convergence.
 
     Returns:
-        (dict) Result dictionary containing:
+        dict: Result dictionary containing:
 
         - solution_x (np.ndarray): Solution vector of shape (n_params,).
         - solution_criterion (np.ndarray): Likelihood contributions at the solution.
@@ -286,12 +286,17 @@ def _estimate_epsilon_inactive_set(x, norm_grad, lower_bounds, upper_bounds):
     The set of epsilon-inactive indices underestimates (overestimates) the actual
     set of inactive (active) indices.
 
+    Args:
         x (np.ndarray): Current parameter vector of shape (n_params,).
         norm_grad (float): Norm of the projected gradient.
         lower_bounds (np.ndarray): 1d array of shape (n_params,) with lower bounds
             for the parameter vector x.
         upper_bounds (np.ndarray): 1d array of shape (n_params,) with upper bounds
-            for the parameter vector x
+            for the parameter vector x/
+
+    Returns:
+        np.ndarray: 1d array of shape (n_inactive_constraints,) containing the set
+            of inactive constraints.
     """
     epsilon = min(np.min(upper_bounds - lower_bounds) / 2, norm_grad)
 
diff --git a/tests/optimization/test_bhhh.py b/tests/optimization/test_bhhh.py
index 9248758dd..1e54eff83 100644
--- a/tests/optimization/test_bhhh.py
+++ b/tests/optimization/test_bhhh.py
@@ -89,7 +89,7 @@ def criterion_and_derivative_logit(x, task="criterion_and_derivative"):
             If task="criterion_and_derivative", compute both.
 
     Returns:
-        (np.ndarray or tuple): If task=="criterion" it returns the output of
+        np.ndarray or tuple: If task=="criterion" it returns the output of
             criterion, which is a 1d numpy array.
             If task=="derivative" it returns the first derivative of criterion,
             which is 2d numpy array.
@@ -121,7 +121,7 @@ def criterion_and_derivative_logit_ibm(x, task="criterion_and_derivative"):
             If task="criterion_and_derivative", compute both.
 
     Returns:
-        (np.ndarray or tuple): If task=="criterion" it returns the output of
+        np.ndarray or tuple: If task=="criterion" it returns the output of
             criterion, which is a 1d numpy array.
             If task=="derivative" it returns the first derivative of criterion,
             which is 2d numpy array.
@@ -153,7 +153,7 @@ def criterion_and_derivative_probit(x, task="criterion_and_derivative"):
             If task="criterion_and_derivative", compute both.
 
     Returns:
-        (np.ndarray or tuple): If task=="criterion" it returns the output of
+        np.ndarray or tuple: If task=="criterion" it returns the output of
             criterion, which is a 1d numpy array.
             If task=="derivative" it returns the first derivative of criterion,
             which is 2d numpy array.
@@ -185,7 +185,7 @@ def criterion_and_derivative_probit_ibm(x, task="criterion_and_derivative"):
             If task="criterion_and_derivative", compute both.
 
     Returns:
-        (np.ndarray or tuple): If task=="criterion" it returns the output of
+        np.ndarray or tuple: If task=="criterion" it returns the output of
             criterion, which is a 1d numpy array.
             If task=="derivative" it returns the first derivative of criterion,
             which is 2d numpy array.
@@ -228,28 +228,6 @@ def result_probit_unconstrained():
     return result.params
 
 
-@pytest.fixture
-def result_logit_constrained():
-    endog, exog = generate_test_data()
-    result = sm.Logit(endog, exog).fit(
-        method="lbfgs", bounds=((-5, 10), (-100, 100), (-100, 100)), disp=False
-    )
-
-    return result.params
-
-
-@pytest.fixture
-def result_probit_constrained():
-    endog, exog = generate_test_data()
-    result = sm.Logit(endog, exog).fit(
-        method="lbfgs",
-        bounds=((-5, np.inf), (-np.inf, np.inf), (-np.inf, np.inf)),
-        disp=False,
-    )
-
-    return result.params
-
-
 @pytest.fixture
 def result_logit_ibm():
     endog, exog = load_ibm_data()
@@ -266,6 +244,16 @@ def result_probit_ibm():
     return result.params
 
 
+@pytest.fixture
+def result_logit_constrained():
+    endog, exog = generate_test_data()
+    result = sm.Logit(endog, exog).fit(
+        method="lbfgs", bounds=((-5, 10), (-100, 100), (-100, 100)), disp=False
+    )
+
+    return result.params
+
+
 # =====================================================================================
 # Tests
 # =====================================================================================
diff --git a/tests/optimization/test_pounders_integration.py b/tests/optimization/test_pounders_integration.py
index a90e763e6..c61921b93 100644
--- a/tests/optimization/test_pounders_integration.py
+++ b/tests/optimization/test_pounders_integration.py
@@ -11,23 +11,6 @@
 from numpy.testing import assert_array_almost_equal as aaae
 
 
-def load_history(start_vec, solver_sub):
-    start_vec_str = np.array2string(
-        start_vec, precision=3, separator=",", suppress_small=False
-    )
-
-    history_x = np.genfromtxt(
-        TEST_FIXTURES_DIR / f"history_x_{start_vec_str}_{solver_sub}_3_8.csv",
-        delimiter=",",
-    )
-    history_criterion = np.genfromtxt(
-        TEST_FIXTURES_DIR / f"history_criterion_{start_vec_str}_{solver_sub}_3_8.csv",
-        delimiter=",",
-    )
-
-    return history_x, history_criterion
-
-
 @pytest.fixture
 def criterion():
     data = pd.read_csv(TEST_FIXTURES_DIR / "pounders_example_data.csv")