  We develop a framework for studying variational problems in Banach spaces
with respect to gradient relations, which encompasses many of the notions of
generalized gradients that appear in the literature. We stress the fact that
our approach is not dependent on function spaces and therefore applies equally
well to functions on metric spaces as to operator algebras. In particular, we
consider analogues of Dirichlet and obstacle problems, as well as first
eigenvalue problems, and formulate conditions for the existence of solutions
and their uniqueness. Moreover, we investigate to what extent a lattice
structure may be introduced on (ordered) Banach spaces via a norm-minimizing
variational problem. A multitude of examples is provided to illustrate the
versatility of our approach.
