Idempotent Interval Analysis and Optimization Problems ∗ G. L. Litvinov (
[email protected]) International Sophus Lie Centre A. N. Sobolevski˘ı(
[email protected]) M. V. Lomonosov Moscow State University 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 sta- tionary 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. Keywords: Idempotent Mathematics, Interval Analysis, idempotent semiring, dis- crete optimization, interval discrete Bellman equation MSC codes: 65G10, 16Y60, 06F05, 08A70, 65K10 Introduction Many problems in the optimization theory and other fields of mathe- matics are nonlinear in the traditional sense but appear to be linear over semirings with idempotent addition.1 This approach is developed systematically as Idempotent Analysis or Idempotent Mathematics (see, e.g., [1]–[8]). In this paper we present an idempotent version of Interval Analysis (its classical version is presented, e.g., in [9]–[12]) and discuss applications of the idempotent matrix algebra to discrete optimization theory. The idempotent interval analysis appears to be best suited for treat- ing problems with order-preserving transformations of input data. It gives exact interval solutions to optimization problems with interval un- arXiv:math/0101080v1 [math.NA] 10 Jan 2001 certainties without any conditions of smallness on uncertainty intervals.