Mathematical Programming

, Volume 57, Issue 1–3, pp 445–458 | Cite as

Optimal and sub-optimal stopping rules for the Multistart algorithm in global optimization

  • B. Betrò
  • F. Schoen


In this paper the problem of stopping the Multistart algorithm for global optimization is considered. The algorithm consists of repeatedly performing local searches from randomly generated starting points. The crucial point in this algorithmic scheme is the development of a stopping criterion; the approach analyzed in this paper consists in stopping the sequential sampling as soon as a measure of the trade-off between the cost of further local searches is greater than the expected benefit, i.e. the possibility of discovering a better optimum.

Stopping rules are thoroughly investigated both from a theoretical point of view and from a computational one via extensive simulation. This latter clearly shows that the simple1-step look ahead rule may achieve surprisingly good results in terms of computational cost vs. final accuracy.

Key words

Global optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. J.O. Berger,Statistical Decision Theory (Springer, Berlin, 1980).Google Scholar
  2. B. Betrò and F. Schoen, “Sequential stopping rules for the Multistart algorithm in global optimisation,”Mathematical Programming 38 (1987) 271–286.Google Scholar
  3. B. Betrò and F. Schoen, “A stochastic technique for global optimization,”Computers and Mathematics with Applications 21 (1991) 127–133.Google Scholar
  4. G. Boender and A.H.G. Rinnooy Kan, “Bayesian stopping rules for multistart global optimization methods,”Mathematical Programming 37 (1987) 59–80.Google Scholar
  5. M.H. DeGroot,Optimal Statistical Decisions (McGraw-Hill, New York, 1970).Google Scholar
  6. W. Feller,An Introduction to Probability Theory and its Applications, Vol. II (Wiley, New York, 1966).Google Scholar
  7. M. Piccioni and A. Ramponi, “Stopping rules for the Multistart method when different local minima have different function values,” to appear in:Optimization (1990).Google Scholar
  8. R. Zielinski, “A statistical estimate of the structure of multi-extremal problem,”Mathematical Programming 21 (1981) 348–356.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1992

Authors and Affiliations

  • B. Betrò
    • 1
  • F. Schoen
    • 2
  1. 1.CNR-IAMIMilanoItaly
  2. 2.Dipartimento di Scienze dell'InformazioneMilanoItaly

Personalised recommendations