I need to find the global minimum of the following polynomial over a four-dimensional sphere.
$$f(x_0, x_1, x_2, x_3) = \left( \sum\limits_{j=0}^3 c_j x_j^2 \right)^2 + \sum\limits_{n,m=0}^3 c_{nm} x_n x_m $$
Here $x_0,x_1,x_2,x_3$ are the variables and are constrained to be on the four-dimensional sphere, i.e.,
$$\sum_{n=0}^3 x_n^2=1$$
and $c_j\ (j = 0, 1, 2, 3)$ and $c_{nm}\ (\{ n, m\} = 0,1,2,3)$ are problem constants that can be positive or negative or zero.
Well, I am aware of the general method; find the derivatives calculate the roots etc... however as you can see the problem has some sort of simple structure (quartic part can be diagonalized) so I am wondering if there is a simpler solution. It is also possible to diagonalize the quadratic part and leave the quartic part complicated.