  This paper provides characterizations of the weak solutions of optimization
problems where a given vector function $F,$ from a decision space $X$ to an
objective space $Y$, is "minimized" on the set of elements $x\in C$ (where
$C\subset X$ is a given nonempty constraint set), satisfying $G\left( x\right)
\leqq_{S}0_{Z},$ where $G$ is another given vector function from $X $ to a
constraint space $Z$ with positive cone $S$. The three spaces $X,Y,$ and $Z$
are locally convex Hausdorff topological vector spaces, with $Y$ and $Z$
partially ordered by two convex cones $K$ and $S,$ respectively, and enlarged
with a greatest and a smallest element. In order to get suitable versions of
the Farkas lemma allowing to obtain optimality conditions expressed in terms of
the data, the triplet $\left( F,G,C\right) ,$ we use non-asymptotic
representations of the $K-$epigraph of the conjugate function of $F+I_{A},$
where $I_{A}$ denotes the indicator function of the feasible set $A,$ that is,
the function associating the zero vector of $Y$ to any element of $A$ and the
greatest element of $Y$ to any element of $X\diagdown A.$
