  In this paper, we study a class of fractional semi-infinite polynomial
programming problems involving s.o.s-convex polynomial functions. For such a
problem, by a conic reformulation proposed in our previous work and the
quadratic modules associated with the index set, a hierarchy of semidefinite
programming (SDP) relaxations can be constructed and convergent upper bounds of
the optimum can be obtained. In this paper, by introducing Lasserre's
measure-based representation of nonnegative polynomials on the index set to the
conic reformulation, we present a new SDP relaxation method for the considered
problem. This method enables us to compute convergent lower bounds of the
optimum and extract approximate minimizers. Moreover, for a set defined by
infinitely many s.o.s-convex polynomial inequalities, we obtain a procedure to
construct a convergent sequence of outer approximations which have semidefinite
representations. The convergence rate of the lower bounds and outer
approximations are also discussed.
