A class of problems for which cyclic relaxation converges linearly

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 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.