Optimization Problem, comparing the steepest descent and grdient descent method

Question

Optimization Problem, comparing the steepest descent and grdient descent method using the data points below and the equation solving for the "a." I mostly need help with the MatLab code since there are many ways to do it:

(Computer problem) Implement a MATLAB routine for implementing the steepest descent algorithm for a quadratic problem. Given the following data set XT -4 -3 -2 -1 0 1 2 T3 T4 T5 T6 y 56 35 21 11 3 1 0.5 6 13 28 48 Apply the MATLAB routine to compute a degree two polynomial f(x a2 aux ao to fit the above data set. Compare this code with the fixed step size method

Explanation / Answer

Code for Constant step size Steepest Descent method
%Functions used by the Algorithm:
function r = f1(xk_yk)
format long g
r = (xk_yk(1)^2 - xk_yk(2))^2 + (xk_yk(1) -1)^2;
function p = gradient_f1(xk_yk)
format long g;
p(1) = 4*(xk_yk(1)^2 - xk_yk(2))*xk_yk(1) + 2*(xk_yk(1) -1) ;
p(2) = -2*(xk_yk(1)^2 - xk_yk(2));
function idx = check_min_t_f1(t1,t2,t3,xk_yk)
[m,idx] = min([f1(xk_yk -t1.*gradient_f1(xk_yk))
f1(xk_yk -t2.*gradient_f1(xk_yk)) f1(xk_yk -t3.*gradient_f1(xk_yk))]);

clear,clf,clc
%Gradient methods f1
clear; echo off;
format long g;
% initial point
x0_y0 = [2 2];
% termination point
e = 0.05; N =1000;
% constant step size
t = 0.15;
xk_yk(1,:) = x0_y0 - t.*gradient_f1(x0_y0);
disp(’iteration
x1
x2
f1(x1,x2)
Norm(gradient)’);
for i = 2:N
xk_yk(i,:) = xk_yk(i-1,:) - t.*gradient_f1(xk_yk(i-1,:));

if f1(xk_yk(i,:)) > f1(xk_yk(i-1,:))
disp(’not a descent direction ... exiting’);
disp(sprintf(’The extremum could not be located after %d iterations’,i));

break;
end
disp(sprintf(’%-4d %3.4f %3.4f %3.4f %3.4f’,i,xk_yk(i,1),xk_yk(i,2),
f1(xk_yk(i,:)),norm(gradient_f1(xk_yk(i,:)))));
if norm(gradient_f1(xk_yk(i,:))) < e
disp(’----------------------------------------------------------------------’);
disp(sprintf(’The minima of f1 found after %d iterations,’,i));
disp(sprintf(’using constant step of %f, gradient methods are
x1=%f x2=%f,’,t, xk_yk(i,1),xk_yk(i,2)));
disp(sprintf(’and the min value of function is =%f’, f1(xk_yk(i,:))));
disp(’----------------------------------------------------------------------’);
break;
end
end

n = 30; %# of contours
format long g
[X,Y] = meshgrid(-1:.02:2,-1.4:.02:4);
Z = (X.^2 - Y).^2 + (X - ones(size(X))).^2;
%Then, generate a contour plot of Z.
[C,h] = contour(X,Y,Z,n);
clabel(C,h),xlabel(’x_1’,’Fontsize’,18),ylabel(’x_2’,’Fontsize’,18),
title(sprintf(’f_1(x_1,x_2),Const Stepsize=%1.3f’,t),’Fontsize’,18);
grid on
hold on;
convergence = [x0_y0’ xk_yk’];
%scatter(convergence(1,:),convergence(2,:));
plot(convergence(1,:),convergence(2,:),’-ro’);

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