Exterior Method

clc;
clear;
% Initialize starting point, function, constraints, weights
syms x y rh rg f g1 g2 g0 F grad_x grad_y
u1 = 2;
u2 = 2;
u3 = 2;
u4 = 2;
% Original function to minimize
f = x^2+y^2-6*x-8*y+10
% First constraint g1 <=0 
g1 = 4*x^2+y^2-16 
% Second constraint g2 <= 0 
g2 = 3*x+5*y-4 
% Non-negativity constraints
g3 = -x
g4 = -y
C = 10E-8;
% Unconstrained function f(x) + u*C*exp(constraint)
    F = f + u1*(C*exp(g1)) + u2*(C*exp(g2)) + u3*(C*exp(g3)) + u4*(C*exp(g4))
% Set initial guess for the two variables x1 as x and x2 as y
    x_val(1) = .5;
    y_val(1) = .5;
% Run for 5 iterations (weights become too large)
for j=1:5
    % Set weights increasing each iteration
    u1 = 10^(j-1)
    u2 = 10^(j-1)
    u3 = 10^(j-1)
    u4 = 10^(j-1)
    % Update function with new weights for the penalty terms
    F = f + (C*exp(g1*u1)) + (C*exp(u2*g2)) + (C*exp(u3*g3)) + (C*exp(u4*g4))
% Gradient of unconstrained function
    grad_x = diff(F,x);
    grad_y = diff(F,y);
% Calculate the partial derivatives
    f1x = diff(grad_x,x);
    f1y = diff(grad_x,y);
    f2x = diff(grad_y,x);
    f2y = diff(grad_y,y);
% Print the gradient 
    partials_initial = [f1x  f1y; f2x f2y];
% Run Newton's algorithm for 10 iterations
for k=1:10
    % Calculate the function values at the current x1 and x2 values
    % Put the two values into a list to be used for matrix multiplication later
    f_new = subs(F,[x,y,u1,u2,u3,u4], [x_val(k),y_val(k),u1,u2,u3,u4]);
    
    grad_x_at = subs(grad_x,[x,y],[x_val(k), y_val(k)]);
    grad_y_at = subs(grad_y,[x,y],[x_val(k), y_val(k)]);
    gradient = [grad_x_at,grad_y_at];
    % Evaluate the partial derivatives at the current x1 and x2 values
    p1x = subs(f1x,[x,y], [x_val(k),y_val(k)]);
    p1y = subs(f1y,[x,y], [x_val(k),y_val(k)]);
    p2x = subs(f2x,[x,y], [x_val(k),y_val(k)]);
    p2y = subs(f2y,[x,y], [x_val(k),y_val(k)]);
    % Put the values in a matrix form and find the inverse matrix
    partials = [p1x  p1y; p2x p2y];
    inv_partials = inv(partials);
    
    % Calculate the deltas be multiplying the inverse the function values from the first step in the loop
    delta(:,k) = -(inv_partials)*gradient';
    
    % Set the new x1 and x2 values by adding the corresponding delta to the previous x1 and x2 value
    x_val(k+1) = x_val(k) + delta(1,k);
    y_val(k+1) = y_val(k) + delta(2,k);
end
    % Print the final x1 and x2 values 
    % Print the final value to check = 0
    x1 = x_val(k+1)
    x2 = y_val(k+1)
    final_value_f1(j) = double(subs(F,[x,y,u1,u2,u3,u4],[x_val(k+1),y_val(k+1),u1,u2,u3,u4]))
end
 final_value_f1 = double(subs(F,[x,y,u1,u2,u3,u4],[x_val(k+1),y_val(k+1),u1,u2,u3,u4]))