Computational Optimization and Applications

, Volume 31, Issue 2, pp 123–143 | Cite as

Quadratic Convergence of a Primal-Dual Interior Point Method for Degenerate Nonlinear Optimization Problems

  • Hiroshi Yamashita
  • Hiroshi Yabe


Recently studies of numerical methods for degenerate nonlinear optimization problems have been attracted much attention. Several authors have discussed convergence properties without the linear independence constraint qualification and/or the strict complementarity condition. In this paper, we are concerned with quadratic convergence property of a primal-dual interior point method, in which Newton’s method is applied to the barrier KKT conditions. We assume that the second order sufficient condition and the linear independence of gradients of equality constraints hold at the solution, and that there exists a solution that satisfies the strict complementarity condition, and that multiplier iterates generated by our method for inequality constraints are uniformly bounded, which relaxes the linear independence constraint qualification. Uniform boundedness of multiplier iterates is satisfied if the Mangasarian-Fromovitz constraint qualification is assumed, for example. By using the stability theorem by Hager and Gowda (1999), and Wright (2001), the distance from the current point to the solution set is related to the residual of the KKT conditions.

By controlling a barrier parameter and adopting a suitable line search procedure, we prove the quadratic convergence of the proposed algorithm.

degenerate nonlinear optimization primal-dual interior point method quadratic convergence 


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© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Mathematical Systems, Inc.TokyoJapan
  2. 2.Department of Mathematical Information ScienceTokyo University of ScienceTokyoJapan

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