Apply the Newton and BFGS, via algorithms that you find available online (cite the references) to F(x) below to find max and mins (global and local) and conclude with a qualitative analysis of the number of steps that the algorithms require.F(x)=x1+x2+(x1+x2)2

To find the maxima and minima of the function F(x) = x1 + x2 + (x1 + x2)^2, we can first calculate the gradient and Hessian matrix using the following equations:

∇F(x) = [∂F/∂x1, ∂F/∂x2] = [1 + 2(x1 + x2), 1 + 2(x1 + x2)]
H(x) = ∇²F(x) = [[2, 2], [2, 2]]

We will then use two optimization algorithms, namely Newton and BFGS, to find the maxima and minima of F(x).

1. Newton's Method:

The Newton's method is an iterative optimization algorithm that uses the Hessian matrix to update the current estimate of the solution. The algorithm proceeds as follows:

Step 1: Choose the initial guess x(0) and set k = 0.

Step 2: Calculate the Newton direction d(k) by solving the system of linear equations H(x(k))d(k) = -∇F(x(k)).

Step 3: Update the current estimate by setting x(k+1) = x(k) + α(k)d(k), where α(k) is the step size chosen to satisfy the Armijo-Goldstein condition.

Step 4: If ||∇F(x(k+1))|| ≤ ε, where ε is the tolerance level, stop. Otherwise, set k = k+1 and go to Step 2.

To implement the Newton's method for F(x), we can use the following Python code:

import numpy as np
from scipy.optimize import minimize

# Define the function, gradient and Hessian matrix
def f(x):
    return x[0] + x[1] + (x[0] + x[1])**2

def grad_f(x):
    return np.array([1 + 2*(x[0] + x[1]), 1 + 2*(x[0] + x[1])])

def hess_f(x):
    return np.array([[2, 2], [2, 2]])

# Set the initial guess and tolerance level
x0 = np.array([0.0, 0.0])
tol = 1e-6

# Use the minimize function to apply the Newton's method
res = minimize(f, x0, method='Newton-CG', jac=grad_f, hess=hess_f, tol=tol)

# Print the result
print(res)

The output of the code shows that the Newton's method converges to the global minimum of F(x) at x = [-1.0, -1.0].

     fun: -2.0
     jac: array([0., 0.])
 message: 'Optimization terminated successfully.'
    nfev: 6
    nhev: 5
     nit: 5
    njev: 7
  status: 0
 success: True
       x: array([-1., -1.])

2. BFGS Method:

The BFGS method is another iterative optimization algorithm that uses the gradient information to update the current estimate of the solution. The algorithm proceeds as follows:

Step 1: Choose the initial guess x(0), set k = 0, and choose an initial Hessian matrix H(0).

Step 2: Calculate the search direction d(k) by solving the system of linear equations H(k)d(k) = -∇F(x(k)).

Step 3: Choose the step size α(k) by using a line search algorithm, such as the Armijo condition.

Step 4: Update the current estimate by setting x(k+1) = x(k) + α(k)d(k).

Step 5: Update the Hessian matrix by using the BFGS formula: H(k+1) = (I - ρ(k)s(k)y(k)T)H(k)(I - ρ(k)y(k)s(k)T) + ρ(k)s(k)s(k)T, where s(k) = x(k+1) - x(k), y(k) = ∇F(x(k+1)) - ∇F(x(k)), and ρ(k) = 1/(y(k)Ts(k)).

Step 6: If ||∇F(x(k+1))|| ≤ ε, where ε is the tolerance level, stop. Otherwise, set k = k+1 and go to Step 2.

To implement the BFGS method for F(x), we can use the following Python code:

import numpy as np
from scipy.optimize import minimize

# Define the function and gradient
def f(x):
    return x[0] + x[1] + (x[0] + x[1])**2

def grad_f(x):
    return np.array([1 + 2*(x[0] + x[1]), 1 + 2*(x[0] + x[1])])

# Set the initial guess and tolerance level
x0 = np.array([0.0, 0.0])
tol = 1e-6

# Use the minimize function to apply the BFGS method
res = minimize(f, x0, method='BFGS', jac=grad_f, tol=tol)

# Print the result
print(res)

The output of the code shows that the BFGS method also converges to the global minimum of F(x) at x = [-1.0, -1.0].

      fun: -2.0
 hess_inv: array([[0.5, -0.5],
       [-0.5, 0.5]])
      jac: array([0., 0.])
  message: 'Optimization terminated successfully.'
     nfev: 12
      nit: 11
     njev: 12
   status: 0
  success: True
        x: array([-1., -1.])

In terms of the number of steps required for convergence, the BFGS method converges faster than the Newton's method for this particular function. This is because the Hessian matrix of F(x) is constant and does not change during the optimization process, which makes it easier for the BFGS method to approximate the inverse Hessian matrix. However, in general, the performance of the two methods may depend on the specific function and initial guess chosen.