  In the setting of real vector spaces, we establish a general set-valued
Ekeland variational principle (briefly, denoted by EVP), where the objective
function is a set-valued map taking values in a real vector space quasi-ordered
by a convex cone $K$ and the perturbation consists of a $K$-convex subset $H$
of the ordering cone $K$ multiplied by the distance function. Here, the
assumption on lower boundedness of the objective function is taken to be the
weakest kind. From the general set-valued EVP, we deduce a number of particular
versions of set-valued EVP, which extend and improve the related results in the
literature. In particular, we give several EVPs for approximately efficient
solutions in set-valued optimization, where a usual assumption for
$K$-boundedness (by scalarization) of the objective function's range is
removed. Moreover, still under the weakest lower boundedness condition, we
present a set-valued EVP, where the objective function is a set-valued map
taking values in a quasi-ordered topological vector space and the perturbation
consists of a $\sigma$-convex subset of the ordering cone multiplied by the
distance function.
