MATLAB: Solving problem for gradient descent

Question

hi,

I am trying to solve the following question using gradient descent method.\

.

I wrote the following code but its giving error.

function [xopt,fopt,niter,gnorm,dx] = grad_descent(varargin)
if nargin==0
    % define starting point

    x0 = [3 3]';
elseif nargin==1
    % if a single input argument is provided, it is a user-defined starting

    % point.

    x0 = varargin{1};
else
    error('Incorrect number of input arguments.')
end
% termination tolerance

tol = 1e-6;
% maximum number of allowed iterations

maxiter = 10;
% minimum allowed perturbation

dxmin = 1e-6;
% step size ( 0.33 causes instability, 0.2 quite accurate)

alpha = 0.1;
% initialize gradient norm, optimization vector, iteration counter, perturbation

gnorm = inf; x = x0; niter = 0; dx = inf;
% define the objective function:

f = @(x1,x2,x3) 4*[x1.^2 + x2-x3].^2 +10;
% plot objective function contours for visualization:

figure(1); clf; ezcontour(f,[-5 5 -5 5]); axis equal; hold on
% redefine objective function syntax for use with optimization:

f2 = @(x) f(x(1),x(2),x(3));
% gradient descent algorithm:

while and(gnorm>=tol, and(niter <= maxiter, dx >= dxmin))
    % calculate gradient:

    g = grad(x);
    gnorm = norm(g);
    % take step:

    xnew = x - alpha*g;
    % check step

    if ~isfinite(xnew)
        display(['Number of iterations: ' num2str(niter)])
        error('x is inf or NaN')
    end
    % plot current point

    plot([x(1) xnew(1)],[x(2) xnew(2)],'ko-')
    refresh
    % update termination metrics

    niter = niter + 1;
    dx = norm(xnew-x);
    x = xnew;
    
end
xopt = x;
fopt = f2(xopt);
niter = niter - 1;
%define the gradient of the objective
% function g = grad(x)
% g = [2*x(1) + x(2)
%     x(1) + 6*x(2)];
function g = grad(x)
g = 4*(x(1).^2 + x(2)-x(3)).^2 +10;

.I saved this code in a file called steepest.m and then I try to run the following command

[xopt,fopt,niter,gnorm,dx]=steepest

.But I get error.

I have actually used the following code (which works) to solve my problem.

function [xopt,fopt,niter,gnorm,dx] = grad_descent(varargin)
if nargin==0
    % define starting point
    x0 = [3 3]';
elseif nargin==1
    % if a single input argument is provided, it is a user-defined starting
    % point.
    x0 = varargin{1};
else
    error('Incorrect number of input arguments.')
end
% termination tolerance
tol = 1e-6;
% maximum number of allowed iterations
maxiter = 10;
% minimum allowed perturbation
dxmin = 1e-6;
% step size ( 0.33 causes instability, 0.2 quite accurate)
alpha = 0.1;
% initialize gradient norm, optimization vector, iteration counter, perturbation
gnorm = inf; x = x0; niter = 0; dx = inf;
% define the objective function:
f = @(x1,x2) x1.^2 + x1.*x2 + 3*x2.^2;
% plot objective function contours for visualization:
figure(1); clf; ezcontour(f,[-5 5 -5 5]); axis equal; hold on
% redefine objective function syntax for use with optimization:
f2 = @(x) f(x(1),x(2));
% gradient descent algorithm:
while and(gnorm>=tol, and(niter <= maxiter, dx >= dxmin))
    % calculate gradient:
    g = grad(x);
    gnorm = norm(g);
    % take step:
    xnew = x - alpha*g;
    % check step
    if ~isfinite(xnew)
        display(['Number of iterations: ' num2str(niter)])
        error('x is inf or NaN')
    end
    % plot current point
    plot([x(1) xnew(1)],[x(2) xnew(2)],'ko-')
    refresh
    % update termination metrics
    niter = niter + 1;
    dx = norm(xnew-x);
    x = xnew;
    
end
xopt = x;
fopt = f2(xopt);
niter = niter - 1;
% define the gradient of the objective
function g = grad(x)
g = [2*x(1) + x(2)
    x(1) + 6*x(2)];

.

This code works perfectly but why my code is not working?

please help

thanks

Answer 1

Hi,

Following code Illustrates the working of Gradient Descent for 3 variables.

To eliminate error changes were made to:

• Initial value
• Maxiter value
• Alpha value

function [xopt,fopt,niter,gnorm,dx] = grad_descent(varargin)
 
 
if nargin==0
    % define starting point
    x0 = [3 3 3]';  
elseif nargin==1
    % if a single input argument is provided, it is a user-defined starting
    % point.
    x0 = varargin{1};
else
    error('Incorrect number of input arguments.')
end
 
% termination tolerance
tol = 1e-6;
 
% maximum number of allowed iterations
maxiter = 100000;
 
% minimum allowed perturbation
dxmin = 1e-6;
 
% step size ( 0.33 causes instability, 0.2 quite accurate)
alpha = 0.000001;
 
% initialize gradient norm, optimization vector, iteration counter, perturbation
gnorm = inf; x = x0; niter = 0; dx = inf;
 
% define the objective function:
f = @(x1,x2,x3) 4*(x1.^2 + x2-x3).^2 +10;  
 
 
% plot objective function contours for visualization:
%figure(1); clf; contour3(f,[-5 5 -5 5 -5 5]); axis equal; hold on
 
% redefine objective function syntax for use with optimization:
f2 = @(x) f(x(1),x(2),x(3));               
 
% gradient descent algorithm:
while and(gnorm>=tol, and(niter <= maxiter, dx >= dxmin))
    % calculate gradient:
    g = grad(x);
    gnorm = norm(g);
    % take step:
    xnew = x - alpha*g;
    % check step
    if ~isfinite(xnew)
        display(['Number of iterations: ' num2str(niter)])
        error('x is inf or NaN')
    end
    % plot current point
    %plot([x(1) xnew(1)],[x(2) xnew(2)],'ko-')
    refresh
    % update termination metrics
    niter = niter + 1;
    dx = norm(xnew-x);
    x = xnew;
    
end
xopt = x;
fopt = f2(xopt);
niter = niter - 1;
 
%define the gradient of the objective
% function g = grad(x)
% g = [2*x(1) + x(2)
%     x(1) + 6*x(2)];
function g = grad(x)                   
g = 4*(x(1).^2 + x(2)-x(3)).^2 +10;

ans =

0.3667

0.3667

0.3667

OR

Alternately make use of the following code for accurate result

fun = @(x) 4*(x(1).^2 + x(2)-x(3)).^2 +10;
x0 = [3,3,3];
x = fminsearch(fun,x0);