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≤i≤n. Our result generalizes several convergence results that have been observed for algorithms applied to gate- and wire-sizing problems that arise in chip design.
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Rautenbach, D., Szegedy, C. A class of problems for which cyclic relaxation converges linearly. Comput Optim Appl 41, 53–60 (2008). https://doi.org/10.1007/s10589-007-9094-0
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DOI: https://doi.org/10.1007/s10589-007-9094-0