Empirical Investigation of the Convergence Speed
of Inclusion Functions in a Global Optimization
Research Group on Artiﬁcial Intelligence of the Hungarian Academy of Sciences and the University
of Szeged, Szeged, Hungary, e-mail: firstname.lastname@example.org
University of Szeged, Institute of Informatics, Szeged, Hungary, e-mail: email@example.com
(Received: 16 February 2004; accepted: 25 November 2004)
Abstract. This paper deals with the empirical convergence speed of inclusion functions applied in
interval methods for global optimization. According to our experience the natural interval extension
of a given function can be as good as a usual quadratically convergent inclusion function, and although
centered forms are in general only of second-order, they can perform as one of larger convergence
order. These facts indicate that the theoretical convergence order should not be the only indicator of
the quality of an inclusion function, it would be better to know which inclusion function can be used
most efﬁciently in concrete instances. For this reason we have investigated the empirical convergence
speed of the usual inclusion functions on some test functions.
The aim of this paper is to analyze the convergence speed of inclusion functions
used in branch and bound algorithms solving global optimization problems . In
such methods inclusion functions play an important role providing bounds on the
range of the objective functions , .
The comparison of inclusion functions is usually done by comparing their
convergence orders. The inclusion function with the higher convergence order
is deemed to be the better function. However, the convergence order only provides
alower bound for the speed, i.e. it shows the worst case. For practical purposes
one would like to know the average behaviour, especially for actually used, ﬁnite
length argument interval sequences. We have therefore made an extensive empirical
investigation measuring the average speed so that we can decide which inclusion
function should be used in a given situation.
This work has been supported by the Grants OTKA T 034350 and T 032118, OMFB D–30/2000,
and OMFB E–24/2001.
The authors are grateful for the anonymous referees for their suggestions.