Minimizing a tri-variate function

Hello!

I have the following task - I need to find the minimum of the following function (using only fminsearch) F(X,Y,Z)=(X-a*Y)^2+(X-a*Z)^2 , by parametre a

The problem is that X, Y, Z are stored in arrays (one dimentional per each variable).

I've written the following code

function F= F_linear(M,F,SP,a) F = (M-a*F)^2 + (M-a*SP)^2; end

But, when I try to use fminsearch (like this:x = fminsearch(@(x) myfun(X,Y,Z,a))) the following error-message occures:

Inputs must be a scalar and a square matrix. To compute elementwise POWER, use POWER (.^) instead.

Error in ==> F_linear at 2 F = (X-a*Y)^2 + (X-a*Z)^2;

Error in ==> @(x)F_linear(X,Y,Z,a)

Error in ==> fminsearch at 191 fv(:,1) = funfcn(x,varargin{:});

How can I fix this problem? Thank you for your reply!

Answer 1

[nick]

Hello terance,

To resolve this error, kindly use element wise operators in the function F_linear, as shown below:

function F = F_linear(M, F, SP, a)
    F = sum((M - a .* F).^2 + (M - a .* SP).^2);
end

You may refer to the documentation by executing the following command in MATLAB Command Window to know more about element-wise operations:

doc power

[Sam Chak]

Hi @nick

Looking at the tri-variate function, the minimum value must occur when .

Since there are two quadratic components, one must solve the equations and .

It is evident that the solutions represent two straight lines. However, I am currently unsure how to implement this in MATLAB and display the results in a 3D projection.

[nick]

Hi @Sam Chak,

The two equations and represent planes in 3D space and for the minimum value of f(x,y,z), these equations gives the following result :

These set of coordinates represents a straight line in the 3D space when and the y-z plane when . The code below shows the intersection region between these 2 equations :

Y = linspace(-10, 10, 100);
Z = linspace(-10, 10, 100);
[Y_grid, Z_grid] = meshgrid(Y, Z);

%  parameter a
a_values = [-2, -1, 0, 1, 2];


figure;
for i = 1:length(a_values)
    a = a_values(i);
    
    % Calculate X for each line
    X_Y = a * Y_grid;
    X_Z = a * Z_grid;
    subplot(2, 3, i);
    hold on;
    
    % Plot the line X = aY
    surf(X_Y, Y_grid, Z_grid, 'FaceAlpha', 0.5, 'EdgeColor', 'none', 'FaceColor', 'r');
    
    % Plot the line X = aZ
    surf(X_Z, Y_grid, Z_grid, 'FaceAlpha', 0.5, 'EdgeColor', 'none', 'FaceColor', 'b');
    xlabel('X');
    ylabel('Y');
    zlabel('Z');
    title(sprintf('a = %.1f', a));
    grid on;
    view(3);
    hold off;
end
sgtitle('3D Projection of Lines X = aY and X = aZ for Different a Values');

figure;

for i = 1:length(a_values)
    a = a_values(i);
    
    subplot(2, 3, i);
    hold on;
    
    if a == 0
        % Special case for a = 0, the solution is the plane X = 0
        X_plane = zeros(size(Y_grid));
        surf(X_plane, Y_grid, Z_grid, 'FaceAlpha', 0.5, 'EdgeColor', 'none', 'FaceColor', 'g');
        title('Solution Plane for a = 0');
    else
        X_line = a * Y;
        Z_line = Y; 
        plot3(X_line, Y, Z_line, 'k', 'LineWidth', 2);
        title(sprintf('a = %.1f', a));
    end
    
    xlabel('X');
    ylabel('Y');
    zlabel('Z');
    grid on;
    view(3);
    hold off;
end



sgtitle('3D Solution Line for X = aY = aZ with Y = Z for Different a Values');