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Computational Optimization and Applications

, Volume 41, Issue 1, pp 53–60 | Cite as

A class of problems for which cyclic relaxation converges linearly

  • Dieter Rautenbach
  • Christian Szegedy
Article

Abstract

The method of cyclic relaxation for the minimization of a function depending on several variables cyclically updates the value of each of the variables to its optimum subject to the condition that the remaining variables are fixed.

We present a simple and transparent proof for the fact that cyclic relaxation converges linearly to an optimum solution when applied to the minimization of functions of the form \(f(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}\frac{x_{i}}{x_{j}}+\sum_{i=1}^{n}(b_{i}x_{i}+\frac{c_{i}}{x_{i}})\) for a i,j ,b i ,c i ∈ℝ≥0 with max {min {b 1,b 2,…,b n },min {c 1,c 2,…,c n }}>0 over the n-dimensional interval [l 1,u 1]×[l 2,u 2⋅⋅⋅×[l n ,u n ] with 0<l i <u i for 1≤in. Our result generalizes several convergence results that have been observed for algorithms applied to gate- and wire-sizing problems that arise in chip design.

Keywords

Cyclic relaxation Coordinate relaxation Method of alternating variables Linear convergence 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institut für MathematikTU IlmenauIlmenauGermany
  2. 2.Cadence Berkeley LabsBerkeleyUSA

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