Reliable Computing 7: 497–505, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
A Surprising Approach in Interval Global
Dedicated to Prof. Dr. Gregory G. Menshikov on the occasion of his 70th
SERGEY P. SHARY
Institute of Computational Technologies, 630090 Novosibirsk, Russia, e-mail: firstname.lastname@example.org
(Received: 2 April 2001; accepted: 10 June 2001)
Abstract. The work advances a new class of global optimization methods, called graph subdivision
methods, that are based on the simultaneous adaptive subdivision of both the function’s domain of
deﬁnition and the range of values.
The subject matter of our communication is the problem of global optimization of
a real-valued function
over an axis-aligned rectangular box X (i.e.
over an interval vector):
The problem (1.1) is known to be (more or less) successfully solved by various
interval techniques , , , which enables one to reliably compute two-sided
bounds for both the optimum value and the argument it is attained at. The basis
of these methods is adaptive, according to the “branch-and-bound” strategy, subdi-
vision of the domain of the function to be minimized combined with the interval
evaluation of the ranges over the resulting subdomains.
The purpose of our work is to present a new promising interval approach for
the solution of the problem (1.1) that relies upon joint adaptive subdivision of
both the function’s domain of deﬁnition and its range of values. For some classes
of problems, the new approach is expected to turn out better than the traditional
techniques from , ,  in either implementation ﬂexibility or computational
efﬁcacy and the quality of the results it produces.
2. Idea of the New Approach
Notice that any function
, being by the very deﬁnition a special
subset of the direct product
,isan(n + 1)-dimensional object. In connection