Installation

To install the METOD Algorithm from source:

$ git clone https://github.com/Megscammell/METOD-Algorithm.git
$ cd METOD-Algorithm
$ python setup.py install

To ensure all tests are working, create an environment and run the tests using pytest:

$ conda env create -f environment.yml
$ conda activate metod_algorithm
$ pytest

An example of applying the METOD Algorithm with an objective function and gradient is presented below:

 >>> import numpy as np
 >>> import math
 >>> import metod_alg as mt
 >>>
 >>> np.random.seed(90)
 >>> d = 2
 >>> A = np.array([[1, 0],[0, 10]])
 >>> theta = np.random.uniform(0, 2 * math.pi)
 >>> rotation = np.array([[math.cos(theta), -math.sin(theta)],
 ...                     [math.sin(theta), math.cos(theta)]])
 >>> x0 = np.array([0.5, 0.2])
 >>>
 >>> def f(x, x0, A, rotation):
 ...     return 0.5 * (x - x0).T @ rotation.T @ A @ rotation @ (x - x0)
 ...
 >>> def g(x, x0, A, rotation):
 ...     return rotation.T @ A @ rotation @ (x - x0)
 ...
 >>> args = (x0, A, rotation)
 >>> (discovered_minimizers,
 ...  number_minimizers,
 ...  func_vals_of_minimizers,
 ...  excessive_no_descents,
 ...  starting_points,
 ...  no_grad_evals) = mt.metod(f, g, args, d, num_points=10)
 >>> assert(np.all(np.round(discovered_minimizers[0], 3) == np.array([0.500, 0.200])))
 >>> assert(number_minimizers == 1)
 >>> assert(np.round(func_vals_of_minimizers, 3) == 0)
 >>> assert(excessive_no_descents == 0)
 >>> assert(np.array(starting_points).shape == (10, d))

The purpose of each line of code within the example is discussed in the following table.

Line number	Purpose of each line of code within the example
1 - 3	Import the required libraries.
5	Initialize the pseudo-random number generator seed.
6	Set the dimension as d = 2.
7	Create the variable A, which is assigned a diagonal matrix.
8	Create the variable theta, which is assigned a value chosen uniformly at random from \([0, 2\pi]\).
9 - 10	Create the variable rotation, which is assigned a rotation matrix using theta.
11	Create the variable x0, which is the minimizer of f.
13 - 14	Define a function f to apply the METOD Algorithm.
16-17	Define the gradient g, which returns the gradient of f.
19	Set x0, A and rotation as objective function arguments. The function arguments are required to run f and g.
20 - 25	Run the METOD Algorithm with f, g, args, d and optional input num_points=10 to obtain the METOD Algorithm outputs.
26 - 30	Check outputs of the METOD Algorithm.