Idempotent interval analysis and optimization problems
Grigori Litvinov, Andrei Sobolevskii
TL;DR
The paper develops an idempotent interval analysis framework for optimization problems that are nonlinear in the traditional sense but linear over idempotent semirings. It introduces set-valued and interval extensions (weak and strong) of idempotent algebra, and establishes exact, polynomial-time outer estimates for interval Bellman equations via the closure $\mathbf A^*\mathbf B$, with a spectral convergence criterion. The results extend to positive semirings, broadening applicability to a wider class of order-preserving problems. This work provides a rigorous, computable approach to interval uncertainty in discrete optimization, enabling robust solution sets and efficient convergence analyses rooted in idempotent linear algebra. Its findings have direct implications for exact interval solutions in interval linear systems and for algorithmic design in graphs and dynamic programming under uncertainty.
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.