The function you have provided is:

$f(x, y) = x^2 + (x - y)^2 + \frac{1}{16}(x + y + 1)^4$

To solve this using the steepest descent method in MATLAB, we follow these general steps:

Outline of the Steepest Descent Method

1. Initialize: Choose an initial guess $(x_0, y_0)$.
2. Compute the Gradient: At each step, compute the gradient of the function $f(x, y)$, denoted by $\nabla f(x, y)$.
3. Update Rule: Update the current point using the formula: $\mathbf{p}_{k+1} = \mathbf{p}_k - \alpha \nabla f(\mathbf{p}_k)$ where $\alpha$ is the step size (learning rate) and $\mathbf{p}_k = (x_k, y_k)$ is the current point.
4. Stopping Criterion: Stop when the gradient becomes sufficiently small or after a certain number of iterations.

Step-by-Step MATLAB Code

matlabCopy code
% Steepest Descent Method for solving f(x, y)

% Define the function
f = @(x, y) x^2 + (x - y)^2 + (1/16) * (x + y + 1)^4;

% Gradient of the function
grad_f = @(x, y) [2*x + 2*(x - y) + (1/4)*(x + y + 1)^3; ...
                 -2*(x - y) + (1/4)*(x + y + 1)^3];

% Parameters for the steepest descent
alpha = 0.01;   % step size (adjust based on convergence speed)
tol = 1e-6;     % tolerance for stopping
max_iter = 1000; % maximum number of iterations

% Initial guess
x0 = 0;  % x initial
y0 = 0;  % y initial

% Steepest Descent Loop
x = x0;
y = y0;
for iter = 1:max_iter
    % Calculate the gradient
    grad = grad_f(x, y);
    
    % Update the points
    x = x - alpha * grad(1);
    y = y - alpha * grad(2);
    
    % Check for convergence (stopping criterion)
    if norm(grad) < tol
        break;
    end
end

% Display the results
fprintf('Optimal x: %f, Optimal y: %f, Iterations: %d\n', x, y, iter);

Explanation:

• f(x, y) defines the function to minimize.
• grad_f(x, y) computes the gradient of $f(x, y)$.
• We start at an initial guess $(x_0, y_0)$ and iteratively update the values of $x$ and $y$ using the steepest descent update rule.
• The loop stops when either the gradient becomes small enough (converges) or the maximum number of iterations is reached.

Steps to Follow:

1. Copy the MATLAB code into your editor.
2. Set your initial values and step size (alpha).
3. Run the code and observe the output to find the minimum of the function.

Would you like further details on specific parts, or do you have any questions?

Related Questions:

1. How do you compute gradients for more complex functions?
2. What are typical methods to choose the step size in steepest descent?
3. How does the steepest descent method compare to other optimization methods, like Newton's method?
4. What if the function $f(x, y)$ had constraints on $x$ and $y$? How would that change the solution?
5. What is the impact of the choice of the initial guess on the convergence of the method?

Tip:

To improve the convergence speed, you can implement a line search to dynamically adjust the step size $\alpha$ at each iteration.