Minimize a function using Newton's Method

Question

Shikhar Singh 2022-4-13

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1695595-minimize-a-function-using-newton-s-method

评论： Matt J 2022-4-13

I am trying to minimise the function stated below using Newton's method, however I am not able to display a plot which illustrates points as they iterate down towards the minimum:

% Minimise z =(3-X).^2 + 30*((Y-(X.^2)).^2) with a starting point of x=y=0

% My implementation:

X = -4:0.01:4; % Declare range of x values

[X,Y] = meshgrid(X); % return 2-D grid coordinates

f =(3-X).^2 + 30*((Y-(X.^2)).^2); % Initiallize function for plotting

surf(X,Y,f,'FaceColor','y','FaceAlpha',0.3,'EdgeColor','none'); % Plot function

syms X Y;

f =(3-X).^2 + 30*((Y-(X.^2)).^2);

x(1) = 0; % Initial value of x

y(1) = 0;% Initial value of y

df_dx = diff(f, X);

df_dy = diff(f, Y);

J = [subs(df_dx,[X,Y], [x(1),y(1)]) subs(df_dy, [X,Y], [x(1),y(1)])]; % Gradient

ddf_ddx = diff(df_dx,X);

ddf_ddy = diff(df_dy,Y);

ddf_dxdy = diff(df_dx,Y);

ddf_ddx_1 = subs(ddf_ddx, [X,Y], [x(1),y(1)]);

ddf_ddy_1 = subs(ddf_ddy, [X,Y], [x(1),y(1)]);

ddf_dxdy_1 = subs(ddf_dxdy, [X,Y], [x(1),y(1)]);

H = [ddf_ddx_1, ddf_dxdy_1; ddf_dxdy_1, ddf_ddy_1]; % Hessian

B = inv(H);

for i = 1:60

A = [x(i),y(i)]';

x(i+1) = A(1)-B(1,:)*J';

y(i+1) = A(2)-B(2,:)*J';

i = i+1;

J = [subs(df_dx,[X,Y], [x(i),y(i)]) subs(df_dy, [X,Y], [x(i),y(i)])]; % Update Jacobian

ddf_ddx_1 = subs(ddf_ddx, [X,Y], [x(i),y(i)]);

ddf_ddy_1 = subs(ddf_ddy, [X,Y], [x(i),y(i)]);

ddf_dxdy_1 = subs(ddf_dxdy, [X,Y], [x(i),y(i)]);

H = [ddf_ddx_1, ddf_dxdy_1; ddf_dxdy_1, ddf_ddy_1]; % Update Hessian

B = inv(H); % New Search Direction

end

% Plot:

hold on;

plot3(x,y,f,'o','Markersize',3,'Color','red'); % Illustrate descent

Error using plot3
Data must be numeric, datetime, duration or an array convertible to double.

xlabel x; ylabel y; zlabel z;

title('Newton Method');

hold off

grid on;

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Answer 1

Matt J 2022-4-13

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1695595-minimize-a-function-using-newton-s-method#answer_941905

Perhaps you should be using the method that you accepted yesterday,

https://www.mathworks.com/matlabcentral/answers/1694790-gradient-descent-for-function-with-multiple-variables?s_tid=srchtitle

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Shikhar Singh 2022-4-13

I have updated the code above with plot3 implemented.

Matt J 2022-4-13

在 MATLAB Online 中打开

You cannot pass symbolic variables (like f) to plot3. In your previous post, you computed a sequence of z values that you fed to plot3,

z(i+1) = ((3-x(i+1)).^2) + (30.*(y(i+1)-(x(i+1).^2)).^2);

You have not done a similar thing here.

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Minimize a function using Newton's Method

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