  We show that the minimization problem of any non-convex and non-lower
semi-continuous function on a compact convex subset of a locally convex real
topological vector space can be studied via an associated convex and lower
semi-continuous function $\Gamma \left( h\right) $. This observation uses the
notion of $\Gamma $-regularization as a key ingredient. As an application we
obtain, on any locally convex real space, a generalization of the Lanford
III-Robinson theorem which has only been proven for separable real Banach
spaces. The latter is a characterization of subdifferentials of convex
continuous functions.
