ABS methods for continuous and integer linear equations and optimization

Spedicato Emilio (), Bodon Elena, Xia Zunquan and Mahdavi-Amiri Nezam

Central European Journal of Operations Research, 2010, vol. 18, issue 1, 73-95

Abstract: ABS methods are a large class of algorithms for solving continuous and integer linear algebraic equations, and nonlinear continuous algebraic equations, with applications to optimization. Recent work by Chinese researchers led by Zunquan Xia has extended these methods also to stochastic, fuzzy and infinite systems, extensions not considered here. The work on ABS methods began almost thirty years. It involved an international collaboration of mathematicians especially from Hungary, England, China and Iran, coordinated by the university of Bergamo. The ABS method are based on the rank reducing matrix update due to Egerváry and can be considered as the most fruitful extension of such technique. They have led to unification of classes of methods for several problems. Moreover they have produced some special algorithms with better complexity than the standard methods. For the linear integer case they have provided the most general polynomial time class of algorithms so far known; such algorithms have been extended to other integer problems, as linear inequalities and LP problems, in over a dozen papers written by Iranian mathematicians led by Nezam Mahdavi-Amiri. ABS methods can be implemented generally in a stable way, techniques existing to enhance their accuracy. Extensive numerical experiments have shown that they can outperform standard methods in several problems. Here we provide a review of their main properties, for linear systems and optimization. We also give the results of numerical experiments on some linear systems. This paper is dedicated to Professor Egerváry, developer of the rank reducing matrix update, that led to ABS methods. Copyright Springer-Verlag 2010

Keywords: Egerváry rank reduction matrix update; ABS methods; Linear systems; Diophantine Linear systems; Quasi-Newton equation; Feasible direction methods for linearly constrained optimization; Simplex method; Primal-dual interior point methods; ABSPACK (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s10100-009-0128-9

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