Standard Article

Ellipsoidal Algorithms

  1. Ricardo H. C. Takahashi

Published Online: 15 FEB 2011

DOI: 10.1002/9780470400531.eorms0295

Wiley Encyclopedia of Operations Research and Management Science

Wiley Encyclopedia of Operations Research and Management Science

How to Cite

Takahashi, R. H. C. 2011. Ellipsoidal Algorithms. Wiley Encyclopedia of Operations Research and Management Science. .

Author Information

  1. Universidade Federal de Minas Gerais, Department of Mathematics, Belo Horizonte, Minas Gerais, Brazil

Publication History

  1. Published Online: 15 FEB 2011

Abstract

This article reviews the ellipsoidal algorithms (EAs), starting from its working principles, and then presenting the basic EA and some deep cut variations in a constructive manner. The classes of problems for which the EAs are guaranteed to converge to the exact solutions (linear problems, convex problems, and quasi-convex problems) are stated in first place. The general principle of cutting-plane branch-and-cut operation, that is valid for such classes of problems, is presented. The basic EA is stated, as an instance of such principles, and the convergence properties of this version of EA are discussed. The basic EA is then reformulated, with the inclusion of the deep cuts, which accelerate the convergence. Specific forms of deep cuts that can be applied to linear- and convex problems are presented. Some further implementation issues in EAs are briefly discussed, and some references are indicated.

Keywords:

  • ellipsoidal algorithm;
  • interior point algorithm;
  • convergence robustness;
  • maximal violated constraint;
  • bounding box;
  • branch-and-cut