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Properties of the log-barrier function on
degenerate nonlinear programs

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Stephen J. Wright and Dominique Orban

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 qualifications 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 obtain a semi-explicit
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.
Preprint ANL/MCS-P772-0799, July, 1999

Contact: wright@mcs.anl.gov