  We derive the dual variational principle (principle of minimal complementary
energy) for the nonlocal nonlinear scalar diffusion problem, which may be
viewed as the nonlocal version of the $p$-Laplacian operator. We establish
existence and uniqueness of solutions (two-point fluxes) as well as their
quantitative stability, which holds uniformly with respect to the small
parameter (nonlocal horizon) characterizing the nonlocality of the problem. We
then focus on the nonlocal analogue of the classical optimal control in the
coefficient problem associated with the dual variational principle, which may
be interpreted as that of optimally distributing a limited amount of
conductivity in order to minimize the complementary energy. We show that this
nonlocal optimal control problem $\Gamma$-converges to its local counterpart,
when the nonlocal horizon vanishes.
