On Interior Logarithmic Smoothing and Strongly Stable Stationary Points

The paper deals with a nonlinear programming problem (P) and, by using a logarithmic barrier function, a parametric family of interior point approximations M-gamma of its feasible set M, where M-gamma is described by a single smooth inequality constraint. Assuming that a stationary point (x) over bar of (P) under consideration is strongly stable, it is shown that for all sufficiently small gamma > 0 there exists locally around (x) over bar a uniquely determined stationary point x(gamma) of (P-gamma), where (P-gamma) is obtained from (P) by substituting M by M-gamma. In particular, x(gamma) is strongly stable, even nondegenerate, and it has the same stationary index as (x) over bar. Furthermore, it turns out that x(gamma) and its uniquely determined Lagrange multiplier mu(gamma) form a solution pair of a corresponding interior-point problem, where (x(gamma), mu(gamma)) depends continuously differentiable (under linear independence constraint qualification (LICQ)) or continuous (under Mangasarian-Fromovitz constraint qualification (MFCQ)) on the parameter. and x(gamma) converges to (x) over bar as gamma -> 0. The stationary point (x) over bar 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-gamma) as well as some further topological properties of M and M. are discussed.