The nonlinear optimization problem (4)-(5)
has as constraints the relations (),
which can be written as .
However, these nonlinear equality constraints are computationally not so easy
to handle as a corresponding reformulation involving spherical coordinates.
With these, we can write every point as

Of course, it is .
Therefore, each point on the three-dimensional unit sphere can be
parametrized by only two variables and
. Our problem now becomes

where

The actual number of unknowns has dropped from 3 *N* to 2 *N*. Moreover,
we have traded the *N* quadratic equality constraints
against simple box constraints
in . On the other hand, the objective function involves now
trigonometric terms, which results in a subsequent higher computational effort
for an objective function evaluation. Nevertheless, our computational tests
with both formulations clearly indicate that the reformulation is favorable.
We may also assume without loss of generality that the facility is fixed
at the north pole,

Moreover, the longitude of can also be fixed:

The *N*-facility problem has now 2 *N* - 3 unknowns. Without fixing at
the north pole and simultaneously fixing the longitude of , every
local minimum can be transformed into another one by an arbitrary rotation of
the facility locations. But even now will arbitrary permutations between the
facility locations produce different optima.

Thu Dec 23 19:39:35 CET 1999