  Recently, Duncan and Mawhinney introduced a method to find saddle points of
the action in simulations of non-abelian lattice gauge theory. The idea, called
`extremization', is to minimize $\int(\delta S/\delta A_\mu)^2$ instead of the
action $S$ itself as in conventional `cooling'. The method was implemented in
an explicitly gauge variant way, however, and gauge dependence showed up in the
results. Here we present a gauge invariant formulaton of extremization on the
lattice, applicable to any gauge group and any lattice action. The procedure is
worked out in detail for U(1) and SU(N) lattice gauge theory with the plaquette
action.
