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
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.