A new approach for the numerical solution of smooth, nonlinear semi-infinite programs whose feasible set contains a nonempty interior is presented. Interval analysis methods are used to construct finite nonlinear, or mixed-integer nonlinear, reformulations of the original semi-infinite program under relatively mild assumptions on the problem structure. In certain cases the finite reformulation is exact and can be solved directly for the global minimum of the semi-infinite program ({SIP}). In the general case, this reformulation is over-constrained relative to the {SIP}, such that solving it yields a guaranteed feasible upper bound to the {SIP} solution. This upper bound can then be refined using a subdivision procedure which is shown to converge to the true {SIP} solution with finite egr-optimality. In particular, the method is shown to converge for {SIPs} which do not satisfy regularity assumptions required by reduction-based methods, and for which certain points in the feasible set are subject to an infinite number of active constraints. Numerical results are presented for a number of problems in the {SIP} literature. The solutions obtained are compared to those identified by reduction-based methods, the relative performances of the nonlinear and mixed-integer nonlinear formulations are studied, and the use of different inclusion functions in the finite reformulation is investigated.