objfcn = @(b,x) b(1).*x.^b(3) + b(2).*x.^b(4);
x = 1:10;
y = rand(1, 10);
B0 = ones(4,1);
[B,rsdnrm] = fminsearch(@(b) norm(y - objfcn(b,x)), B0);
fprintf(1, 'c_1 = %12.6f\nc_2 = %12.6f\nn_1 = %12.6f\nn_2 = %12.6f\n', B)
Experiment to get the result you want.
EDIT — (24 Sep 2019 at 15:19)
I have no idea how ‘A’ fits into this, since it was not described in any detail.
[B,rsdnrm] = fminsearch(@(b) norm(A(:,2) - objfcn(b,A(:,1))), B0);
This fits the second column of ‘A’ to the first column, using the function. Reverse the column designations if necessary.