  Extremal problems are studied involving an objective function with values in
(order) complete lattices of sets generated by so called set relations.
Contrary to the popular paradigm in vector optimization, the solution concept
for such problems, introduced by F. Heyde and A. L\"ohne, comprises the
attainment of the infimum as well as a minimality property. The main result is
a Minty type variational inequality for set optimization problems which
provides a sufficient optimality condition under lower semicontinuity
assumptions and a necessary condition under appropriate generalized convexity
assumptions. The variational inequality is based on a new Dini directional
derivative for set-valued functions which is defined in terms of a "lattice
difference quotient": A residual operation in a lattice of sets replaces the
inverse addition in linear spaces. Relationships to families of scalar problems
are pointed out and used for proofs: The appearance of improper scalarizations
poses a major difficulty which is dealt with by extending known scalar results
such as Diewert's theorem to improper functions.
