4 views (last 30 days)

clc

clear

format long

% Function Definition (Enter your Function here):

syms X Y;

f = 100*((Y-X^2)^2+(1-X))^2;

% Initial Guess:

x(1) = -0.75;

y(1) = 1;

e = 10^(-8); % Convergence Criteria

i = 1; % Iteration Counter

% Gradient Computation:

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

S = -(J); % Search Direction

% Minimization Condition:

while norm(J) > e

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

syms h; % Step size

g = subs(f, [X,Y], [x(i)+S(1)*h,y(i)+h*S(2)]);

dg_dh = diff(g,h);

h = solve(dg_dh, h); % Optimal Step Length

x(i+1) = I(1)+h(1)*S(1); % Updated x value

y(i+1) = I(2)+h(2)*S(2); % Updated y value

i = i+1;

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

S = -(J); % New Search Direction

end

% Result Table:

Iter = 1:i;

X_coordinate = x';

Y_coordinate = y';

Iterations = Iter';

T = table(Iterations,X_coordinate,Y_coordinate);

% Plots:

contour3(f, 'Fill', 'On');

hold on;

plot(x,y,'*-r');

% Output:

print('Initial Objective Function Value: %d\n\n',subs(f,[X,Y], [x(1),y(1)]));

if (norm(J) < e)

print('Minimum succesfully obtained...\n\n');

end

print('Number of Iterations for Convergence: %d\n\n', i);

print('Point of Minima: [%d,%d]\n\n', x(i), y(i));

print('Objective Function Minimum Value Post-Optimization: %d\n\n', subs(f,[X,Y], [x(i),y(i)]));

disp(T);

Walter Roberson
on 21 Mar 2020

Edited: Walter Roberson
on 21 Mar 2020

f = 100*((Y-X^2)^2+(1-X))^2;

That is degree 8 in X.

g = subs(f, [X,Y], [x(i)+S(1)*h,y(i)+h*S(2)]);

That substitutes h linearly in X, so the degree of h will be the same as the maximum of the degree of X and Y, so g will end up degree 8 in h.

dg_dh = diff(g,h);

differentiate degree 8 to get degree 7. Which happens to be factorable into degree 4 and degree 3.

h = solve(dg_dh, h); % Optimal Step Length

Degree 7 so you get 7 solutions, some of which are complex valued.

x(i+1) = I(1)+h(1)*S(1); % Updated x value

y(i+1) = I(2)+h(2)*S(2); % Updated y value

The solutions are ordered, but they are ordered according to internal logic that is not documented anywhere. They are certainly not sorted by anything having to do with x and y. The branches associated with the order is going to change as the values change, and at some point it is going to happen to select complex branches according to the internal logic.

You need to think explicitly about how you are going to deal with those multiple roots.

I would suggest that you should be using different symbolic h for x and y directions, calculating the gradient, solving for the two different h variables. If you do this, you will get 8 possible values each for the two different variables. You can then use an algorithm to choose particular ones, such as eliminating the ones with complex solutions, and choosing the solution pair that has the highest sum-of-squared values.

Walter Roberson
on 21 Mar 2020

clc

clear

format long

% Function Definition (Enter your Function here):

syms X Y;

f = 100*((Y-X^2)^2+(1-X))^2;

% Initial Guess:

x(1) = -0.75;

y(1) = 1;

e = 10^(-8); % Convergence Criteria

i = 1; % Iteration Counter

% Gradient Computation:

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

S = -(J); % Search Direction

% Minimization Condition:

syms h k; % Step size

while norm(J) > e

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

g = subs(f, [X,Y], [x(i)+S(1)*h,y(i)+k*S(2)]);

dg_dh = diff(g,h);

dg_dk = diff(g,k);

sol = solve([dg_dh, dg_dk], [h, k]) ;

%expect about 8 solutions, some of which will be complex valued

H = double(sol.h);

K = double(sol.k);

%get rid of the complex-valued ones

mask = imag(H) == 0 & imag(K) == 0;

H = H(mask);

K = K(mask);

%now you still have to pick one, but which? Arbitrarily take the pair

%with the greatest sum-of-squares

[~, maxidx] = max(H.^2 + K.^2);

H = H(maxidx);

K = K(maxidx);

x(i+1) = I(1)+H*S(1); % Updated x value

y(i+1) = I(2)+K*S(2); % Updated y value

i = i+1;

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

S = -(J); % New Search Direction

end

% Result Table:

Iter = 1:i;

X_coordinate = x';

Y_coordinate = y';

Iterations = Iter';

T = table(Iterations,X_coordinate,Y_coordinate);

% Plots:

M = 1;

xmin = min([-M; X_coordinate]);

xmax = max([M; X_coordinate]);

ymin = min([-M; Y_coordinate]);

ymax = max([M; Y_coordinate]);

xvec = linspace(xmin,xmax,50);

yvec = linspace(ymin,ymax,51);

fval = double(subs(subs(f, X, xvec), Y, yvec.'));

contour3(xvec, yvec, fval, 'Fill', 'On');

hold on;

plot(x,y,'*-r');

% Output:

fprintf('Initial Objective Function Value: %d\n\n',subs(f,[X,Y], [x(1),y(1)]));

if (norm(J) < e)

fprintf('Minimum succesfully obtained...\n\n');

end

fprintf('Number of Iterations for Convergence: %d\n\n', i);

fprintf('Point of Minima: [%d,%d]\n\n', x(i), y(i));

fprintf('Objective Function Minimum Value Post-Optimization: %d\n\n', subs(f,[X,Y], [x(i),y(i)]));

disp(T);

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⋮## Direct link to this comment

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