  We provide new sufficient conditions for the finiteness of the optimal value
and existence of solutions to a general problem of minimizing a proper closed
function over a nonempty closed set. The conditions require an asymptotically
bounded decay of a function, a relaxation of p-supercoercivity, and a certain
relation for the asymptotic cone of the constraint set and the asymptotic
function of the objective function. Our analysis combines these conditions with
a regularization technique. We refine the notion of retractive directions of a
set, extend its definition to functions, and establish some basic relations for
such directions for both sets and functions. Using these tools, we provide
existence of solutions results that generalize many of the results in the
literature for both non-convex and convex problems.
