  In this paper, we study a class of fractional semi-infinite polynomial
programming (FSIPP) problems, in which the objective is a fraction of a convex
polynomial and a concave polynomial, and the constraints consist of infinitely
many convex polynomial inequalities. To solve such a problem, we first
reformulate it to a pair of primal and dual conic optimization problems, which
reduce to semidefinite programming (SDP) problems if we can bring
sum-of-squares structures into the conic constraints. To this end, we provide a
characteristic cone constraint qualification for convex semi-infinite
programming problems to guarantee strong duality and also the attainment of the
solution in the dual problem, which is of its own interest. In this framework,
we first present a hierarchy of SDP relaxations with asymptotic convergence for
the FSIPP problem whose index set is defined by finitely many polynomial
inequalities. Next, we study four cases of the FSIPP problems which can be
reduced to either a single SDP problem or a finite sequence of SDP problems,
where at least one minimizer can be extracted. Then, we apply this approach to
the four corresponding multi-objective cases to find efficient solutions.
