  We show that the ``classical'' Harder-Narasimhan filtration associated to a
non semistable vector bundle $E$ can be viewed as a limit object for the action
of the gauge group in the direction of an optimal destabilizing vector. This
vector appears as an extremal value of the so called "maximal weight function".
We give a complete description of these optimal destabilizing endomorphisms.
Then we show that this principle holds for another important moduli problem:
holomorphic pairs (i.e. holomorphic vector bundles coupled with morphisms with
fixed source). We get a generalization of the Harder-Narasimhan filtration
theorem for the associated notion of $\tau $-stability. These results suggest
that the principle holds in the whole gauge theoretical framework.
