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Properties of the Log-Barrier Function on Degenerate Nonlinear Programs

Stephen J. Wright (swright***at***cs.wisc.edu)
Dominique Orban (orban***at***ece.northwestern.edu)

Abstract: 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.

Keywords: Interior-point method, degenerate programs, Mangasarian and Fromovitz constraint qualification, strict complementarity

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation: CERFACS Report TR/PA/99/36 CERFACS - 42, Avenue Gaspard Coriolis 31057 Toulouse Cedex 1, FRANCE July 1999. Revised May 2001 and January, 2002. Published in {\em Mathematics of Operations Research} 27 (2002), pp. 585--613.

Download: [Postscript][PDF]

Entry Submitted: 06/01/2001
Entry Accepted: 06/01/2001
Entry Last Modified: 05/29/2003

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