Reliable Computing 7: 353–377, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
Idempotent Interval Analysis and Optimization
GRIGORI L. LITVINOV
International Sophus Lie Centre, Nagornaya, 27-4-72, Moscow 113186, Russia,
ANDREI N. SOBOLEVSKI
M. V. Lomonosov Moscow State University, Russia, e-mail: email@example.com
(Received: 27 January 2000; accepted: 10 September 2000)
Abstract. Many problems in optimization theory are strongly nonlinear in the traditional sense
but possess a hidden linear structure over suitable idempotent semirings. After an overview of
“Idempotent Mathematics” with an emphasis on matrix theory, interval analysis over idempotent
semirings is developed. The theory is applied to construction of exact interval solutions to the interval
discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the
traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the
case of positive semirings is outlined.
Many problems in the optimization theory and other ﬁelds of mathematics are non-
linear in the traditional sense but appear to be linear over semirings with idempotent
This approach is developed systematically as Idempotent Analysis or
Idempotent Mathematics (see, e.g., , , , , , , , ). In this
paper we present an idempotent version of Interval Analysis (its classical version is
presented, e.g., in , , , ) and discuss applications of the idempotent
matrix algebra to discrete optimization theory.
The idempotent interval analysis appears to be best suited for treating problems
with order-preserving transformations of input data. It gives exact interval solu-
tions to optimization problems with interval uncertainties without any conditions
of smallness on uncertainty intervals. Solution of an interval system is typically
NP-hard in the traditional interval linear algebra; in the idempotent case it is poly-
nomial. The idempotent interval analysis is particularly effective in problems that
are strongly nonlinear in the traditional sense but possess a hidden linear structure
The work was supported by the RFBR grant No. 99-01-01198 and the Erwin Schr
Institute for Mathematical Physics.
One of the most important examples of an idempotent semiring is the set
= + (see Example 1.1 in Subsection 1.2), sometimes called the max-plus