Properties of the log-barrier function on degenerate nonlinear programs

We examine the sequence of local minimizers of the log-barrier function for a nonlinear program near a solution at which second-order sufficient conditions and the Mangasarian–Fromovitz constraint qualification are satisfied, but the active constraint gradients are not necessarily linearly independent. When a strict complementarity condition is satisfied, we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution, and we obtain a semiexplicit characterization of this point. When strict complementarity does not hold, we obtain several other interesting characterizations, in particular, an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter, and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution.