  In this paper, we provide a new scheme for approximating the weakly efficient
solution set for a class of vector optimization problems with rational
objectives over a feasible set defined by finitely many polynomial
inequalities. More precisely, we present a procedure to obtain a sequence of
explicit approximations of the weakly efficient solution set of the problem in
question. Each approximation is the intersection of the sublevel set of a
single polynomial and the feasible set. To this end, we make use of the
achievement function associated with the considered problem and construct
polynomial approximations of it over the feasible set from above. Remarkably,
the construction can be converted to semidefinite programming problems. Several
nontrivial examples are designed to illustrate the proposed new scheme.
