Applies the BFGS algorithm to minimize a differentiable function.

Performs unconstrained minimization of a differentiable function using the BFGS scheme. For details of the algorithm, see [Nocedal and Wright(2006)][1].


The following example demonstrates the BFGS optimizer attempting to find the minimum for a simple two dimensional quadratic objective function.

  minimum = np.array([1.0, 1.0])  # The center of the quadratic bowl.
  scales = np.array([2.0, 3.0])  # The scales along the two axes.

  # The objective function and the gradient.
  def quadratic(x):
    value = tf.reduce_sum(scales * (x - minimum) ** 2)
    return value, tf.gradients(value, x)[0]

  start = tf.constant([0.6, 0.8])  # Starting point for the search.
  optim_results = tfp.optimizer.bfgs_minimize(
      quadratic, initial_position=start, tolerance=1e-8)

  with tf.Session() as session:
    results =
    # Check that the search converged
    # Check that the argmin is close to the actual value.
    np.testing.assert_allclose(results.position, minimum)
    # Print out the total number of function evaluations it took. Should be 6.
    print ("Function evaluations: %d" % results.num_objective_evaluations)


[1]: Jorge Nocedal, Stephen Wright. Numerical Optimization. Springer Series in Operations Research. pp 136-140. 2006


  • value_and_gradients_function: A Python callable that accepts a point as a real Tensor and returns a tuple of Tensors of real dtype containing the value of the function and its gradient at that point. The function to be minimized. The first component of the return value should be a real scalar Tensor. The second component (the gradient) should have the same shape as the input value to the function.
  • initial_position: Tensor of real dtype. The starting point of the search procedure. Should be a point at which the function value and the gradient norm are finite.
  • tolerance: Scalar Tensor of real dtype. Specifies the gradient tolerance for the procedure. If the supremum norm of the gradient vector is below this number, the algorithm is stopped.
  • x_tolerance: Scalar Tensor of real dtype. If the absolute change in the position between one iteration and the next is smaller than this number, the algorithm is stopped.
  • f_relative_tolerance: Scalar Tensor of real dtype. If the relative change in the objective value between one iteration and the next is smaller than this value, the algorithm is stopped.
  • initial_inverse_hessian_estimate: Optional Tensor of the same dtype as the components of the output of the value_and_gradients_function. If specified, the shape should be initial_position.shape * 2. For example, if the shape of initial_position is [n], then the acceptable shape of initial_inverse_hessian_estimate is as a square matrix of shape [n, n]. If the shape of initial_position is [n, m], then the required shape is [n, m, n, m]. For the correctness of the algorithm, it is required that this parameter be symmetric and positive definite. Specifies the starting estimate for the inverse of the Hessian at the initial point. If not specified, the identity matrix is used as the starting estimate for the inverse Hessian.
  • max_iterations: Scalar positive int32 Tensor. The maximum number of iterations for BFGS updates.
  • parallel_iterations: Positive integer. The number of iterations allowed to run in parallel.
  • name: (Optional) Python str. The name prefixed to the ops created by this function. If not supplied, the default name 'minimize' is used.


  • optimizer_results: A namedtuple containing the following items:
  • converged: Scalar boolean tensor indicating whether the minimum was found within tolerance.
  • failed: Scalar boolean tensor indicating whether a line search step failed to find a suitable step size satisfying Wolfe conditions. In the absence of any constraints on the number of objective evaluations permitted, this value will be the complement of converged. However, if there is a constraint and the search stopped due to available evaluations being exhausted, both failed and converged will be simultaneously False.
  • num_objective_evaluations: The total number of objective evaluations performed.
  • position: A tensor containing the last argument value found during the search. If the search converged, then this value is the argmin of the objective function.
  • objective_value: A tensor containing the value of the objective function at the position. If the search converged, then this is the (local) minimum of the objective function.
  • objective_gradient: A tensor containing the gradient of the objective function at the position. If the search converged the max-norm of this tensor should be below the tolerance.
  • inverse_hessian_estimate: A tensor containing the inverse of the estimated Hessian.