Mathematical Programming

, Volume 24, Issue 1, pp 177–215

Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods

  • Masakazu Kojima
  • Yoshitsugu Yamamoto

DOI: 10.1007/BF01585103

Cite this article as:
Kojima, M. & Yamamoto, Y. Mathematical Programming (1982) 24: 177. doi:10.1007/BF01585103


In this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by Van der Laan and Talman. We introduce a new concept ‘primal—dual pair of subdivided manifolds’ and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms. Our basic model furnishes interpretations to several existing methods: Lemke's algorithm for the linear complementarity problem, its extension to the nonlinear complementarity problem, a variable dimension algorithm on conical subdivisions and Merrill's algorithm. We shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us. A method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed.

Key words

Variable Dimension Algorithm Fixed Point Subdivided Manifold Nonlinear Equations 

Copyright information

© The Mathematical Programming Society, Inc. 1982

Authors and Affiliations

  • Masakazu Kojima
    • 1
  • Yoshitsugu Yamamoto
    • 2
  1. 1.Department of Information SciencesTokyo Institute of TechnologyTokyoJapan
  2. 2.Institute of Socio-Economic PlanningUniversity of TsukubaIbarakiJapan

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