In this paper, we consider a class of optimization problems
of minimizing a (at least) twice continuously di erentiable function (probably nonconvex)
f(x) : Rn ! R over a product of multiple balls/spheres constraints. Upon
rescaling the balls/spheres, we cast without loss of generality such class of minimization
problems in the following form:
(BCOP)
8><
>:
min
x2Rn
f(x)
s:t: ci(x) := kx[i]k2 1 = 0; i 2 E;
ci(x) := kx[i]k2 1 0; i 2 I;
where E = f1; 2; : : : ;m1g, I = fm1 + 1;m1 + 2; : : : ;mg, x[i] 2 Rpi , x = (xT
[1]; xT
[2]; : : : ;
xT
[m])T , n =
Pm
i=1 pi, and k k stands for the `2 vector norm. Here, we introduce
the notation x[i] 2 Rpi to represent the ith subvector of x 2 Rn and formulate the
product of multiple ball/sphere constraints as a set of equality and inequality constraints.
To simplify subsequent presentation, we call the above programming the
ball/sphere constrained optimization problem (BCOP). We emphasize that this problem
does not allow overlap among the variables x[i] and therefore the constraints are
separable. However, these variables x[i] may be linked together through the objective
function f(x).