Question

Construct a MATLAB/EXCEL program that will perform a "grid search" on the region from 0< x<10 and 0y 10. Use this grid search to find the (x,y) that minimizes the following function: 28y3 97x2295y2 264x 1386y 577 f(x, y) x4y - 16x3.

Answer 1

clear
clc

%% anonymously defining f(x,y) for easy usage in the entire program
f = @(x,y) x^4 + y^4 - 16*x^3 - 28*y^3 + 97*x^2 + 295*y^2 - 264*x -1386*y + 577;

%% defining the number of points or length (len) for x and y and the equally spaced values using linspace in the range [0 10]
xlen = 100; ylen = 100; % the larger the len the more accurate the results
x = linspace(0,10,xlen);
y = linspace(0,10,ylen);

%% initialising the solution xmin and ymin with the largest values possible and computing fmin at those values
xmin = max(x);
ymin = max(y);
fmin = f(xmin,ymin);

%% grid searching where every possible combination of x and y values is used to compute the value of f(x,y)
for i=1:xlen
for j=1:ylen
fvalue = f(x(i),y(j));
if fvalue<fmin % ensuring the search is kept moving towards the minimum direction
fmin = fvalue; %updating fmin with the current minimum value
xmin = x(i); ymin = y(j); %updating xmin and ymin
end
end
end

%% displaying results
disp('OPTIMUM RESULTS')
fprintf(['fmin: ' num2str(fmin) '\n'])
fprintf(['xmin: ' num2str(xmin) '\n'])
fprintf(['ymin: ' num2str(ymin) '\n'])