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


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.


Cyclic relaxation Coordinate relaxation Method of alternating variables Linear convergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming—Theory and Algorithms. Wiley, New York (1993) MATHGoogle Scholar
  2. 2.
    Chen, C.-P., Chu, C.C.N., Wong, D.F.: Fast and exact simultaneous gate and wire sizing by Lagrangian relaxation. In: Proceedings of the 1998 IEEE/ACM International Conference on Computer-Aided Design, pp. 617–624 (1998) Google Scholar
  3. 3.
    Chu, C.C.N., Wong, D.F.: Greedy wire-sizing is linear time. In: Proceedings of the 1998 International Symposium on Physical Design, pp. 39–44 (1998) Google Scholar
  4. 4.
    Cong, J., He, L.: Local-refinement-based optimization with application to device and interconnect sizing. IEEE Trans. Comput.-Aided Design 18, 406–420 (1999) CrossRefGoogle Scholar
  5. 5.
    Langkau, K.: Gate-sizing in VLSI-design. Diploma thesis, University of Bonn, Germany (2000), 78 pp. (in German) Google Scholar
  6. 6.
    Luo, Z.Q., Tseng, P.: On the convergence of the coordinate descent method for convex differentiable minimization. J. Optim. Theory Appl. 72, 7–35 (1992) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Minoux, M.: Mathematical Programming: Theory and Algorithms. Wiley, Chichester (1986) MATHGoogle Scholar
  8. 8.
    Polyak, B.T.: A general method of solving extremum problems. Dokl. Akad. Nauk SSSR 174(1), 33–36 (1967) (in Russian). Translated in Sov. Math. Dokl. 8(3) 593–597 (1967) MathSciNetGoogle Scholar
  9. 9.
    Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9, 14–29 (1969) CrossRefGoogle Scholar
  10. 10.
    Sechen, C., Tennakoon, H.: Gate sizing using Lagrangian relaxation combined with a fast gradient-based pre-processing step. In: Proceedings of the 2002 IEEE/ACM International Conference on Computer-Aided Design, pp. 395–402 (2002) Google Scholar
  11. 11.
    Sutti, C.: Remarks on the convergence of cyclic coordinate methods. Riv. Mat. Univ. Parma IV Ser. 9, 167–172 (1983) MATHMathSciNetGoogle Scholar
  12. 12.
    Tseng, P.: Convergence of block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109, 475–494 (2001) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Yin, X.: Global convergence of the coordinate relaxation method for nongradient mappings. Commun. Appl. Nonlinear Anal. 13, 85–96 (2006) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

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

Personalised recommendations