Mathematical Programming

, Volume 86, Issue 3, pp 615–635 | Cite as

Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumption

  • Matthias Heinkenschloss
  • Michael Ulbrich
  • Stefan Ulbrich

Abstract.

A class of affine-scaling interior-point methods for bound constrained optimization problems is introduced which are locally q–superlinear or q–quadratic convergent. It is assumed that the strong second order sufficient optimality conditions at the solution are satisfied, but strict complementarity is not required. The methods are modifications of the affine-scaling interior-point Newton methods introduced by T. F. Coleman and Y. Li (Math. Programming, 67, 189–224, 1994). There are two modifications. One is a modification of the scaling matrix, the other one is the use of a projection of the step to maintain strict feasibility rather than a simple scaling of the step. A comprehensive local convergence analysis is given. A simple example is presented to illustrate the pitfalls of the original approach by Coleman and Li in the degenerate case and to demonstrate the performance of the fast converging modifications developed in this paper.

Key words: bound constraints – affine scaling – interior-point algorithms – superlinear convergence – nonlinear programming – degeneracy – optimality conditions Mathematics Subject Classification (1991): 49M05, 49M15, 49M37, 90C06 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Matthias Heinkenschloss
    • 1
  • Michael Ulbrich
    • 2
  • Stefan Ulbrich
    • 2
  1. 1.Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77251–1892, USA, e-mail: heinken@caam.rice.edu.US
  2. 2.Zentrum Mathematik, Technische Universität München, 80290 München, Germany, e-mail: mulbrich/sulbrich@mathematik.tu-muenchen.de.DE

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