What kind of optimization problem is this? Is it a valid problem at all?

Question

minimize $-r_1*r_2*...*r_n$
Equation constraint: $x_1^2/r_1^2 +... + x_n^2/r_n^2=1$
m nonlinear inequality constraint : $f_i(x_1,...,x_n) \geq 0$ $i=1,...,m$

I want to find the ellipsoid with maximal volume. The constraints limit the size of the ellipsoid, they are functions with n variable.