The paper deals with a nonlinear programming problem (P) and, by using a logarithmic barrier function, a parametric family of interior point approximations of its feasible set M, where is described by a single smooth inequality constraint. Assuming that a stationary point of (P) under consideration is strongly stable, it is shown that for all sufficiently small there exists locally around a uniquely determined stationary point of (P), where (P) is obtained from (P) by substituting M by . In particular, is strongly stable, even nondegenerate, and it has the same stationary index as . Furthermore, it turns out that and its uniquely determined Lagrange multiplier form a solution pair of a corresponding interior-point problem, where depends continuously differentiable (under linear independence constraint qualification (LICQ)) or continuous (under Mangasarian–Fromovitz constraint qualification (MFCQ)) on the parameter and converges to as . The stationary point might be degenerate and, a priori, no strict complementarity is assumed. Finally, a globalization of this one-to-one correspondence between the stationary points of (P) and (P) as well as some further topological properties of M and are discussed.