steepest descent using armijo rule

I need help using the armijo rule to find the steepest descent.

Here is the proplem I need to solve and the code I have so far. I know how to do it in 1-D but am having trouble translating that to 2-D.

Code a function to perform a generic steepest descent algorithm using the Armijo line-search rule. Your function should take as inputs, the number of iterations, the function to be minimized (fm), another function that returns the gradient of fm, some initial point x0, and the parameters needed for the line search. Turn in all of your code and the first five iterations when the algorithm is applied to minimizing the function

f(x,y)=(x-2y^2)(x-3y^2), starting at the initial point (-1,-1)

Use line-search parameters of s=1, beta=1/2, and sigma=0.1.

f=@(x) (x(1)-2*x(2).^2).*(x(1)-3*x(2).^2)

xk=[-1,-1]';

fg=@(x) [(x(1)-3*x(2).^2)+(x(1)-2*x(2).^2);4*x(2)*(x(1)-3*x(2).^2)+6*x(2)*(x(1)-2*x(2).^2)];

s=1;

beta=0.5;

sigma=0.1;

for j=1:6

xk(j+1)=armijostep(f,fg,-sign(fg(xk(j))),xk(j),s,beta,sigma); %inputs are f, fg, d, xk, s, beta, sigma in that order

end

hold on;

plot(xk,f(xk),'k*-')

xk

hold off;

[Torsten]

Use xk{j} instead of xk(j) if xk is n-dimensional (n>1) (already set the inital xk as xk{1} = [-1,-1].')

[Kendall Donahoe]

I just tried that. I don't think it did anything. I'm not sure if I did it right. Before my main issue was getting an error on the number of array elements when trying to enter the gradient.

Answer 1

[Brahmadev]

Hi @Kendall Donahoe,

Assuming that 'armijostep' function written by you takes the x and y co-ordinates (2x1 vector) from the current iteration and returns the x and y co-ordinates for the next iteration as a 2x1 vector, we can store pre-allocate 'xk' and modify the code as shown below:

% Defining the function f
f=@(x) (x(1)-2*x(2).^2).*(x(1)-3*x(2).^2);
no_of_iter = 6; % Defining no. of iterations
xk =  zeros([2, no_of_iter]); % Pre-allocating for each step
xk(:, 1)=[-1,-1]'; % Initial guess
% Defining the function fg
fg=@(x) [(x(1)-3*x(2).^2)+(x(1)-2*x(2).^2);4*x(2)*(x(1)-3*x(2).^2)+6*x(2)*(x(1)-2*x(2).^2)];

% Defining constants
s=1;
beta=0.5;
sigma=0.1;

% Calculating each new iteration and plotting it
hold on
for j=1:no_of_iter
    xk(:, j+1) = armijostep(f, fg, -sign(fg(xk(:, j))), xk(:, j), s, beta, sigma); %inputs are f, fg, d, xk, s, beta, sigma in that order
    plot(xk(:, j),f(xk(:, j)),'k*-'); % Plotting each iteration
end
hold off

Hope this helps in resolving your query!