This can be done nicely in a least squares approach - especially with Matlab mldivide operator.
I will give you an example
% Generate Test Data
x1 = rand(10,1);
x2 = rand(10,1);
y = x1 + 2*x2;
% Solve for w1 and w2
w = [x1 x2]\y
If you have additional conditions such as w2 = 1 - w1 (x1+x2=1) just add those to your equation and solve
f.ex. y = w1*x1 + w2*x with w2 = 1 - w1
y = w1x1 +(1-w1)x2
y - x2 = y_hat = w1 x_hat = w1 (x1-x2)
And then use the first solution
