A class of problems for which cyclic relaxation converges linearly

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 ai,j,bi,ci∈ℝ≥0 with max {min {b1,b2,…,bn},min {c1,c2,…,cn}}>0 over the n-dimensional interval [l1,u1]×[l2,u2]×⋅⋅⋅×[ln,un] with 0<li<ui 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.