Computing Bounds for Imprecise Continuous-Time Markov Chains Using Normal Cones

Abstract

The theory of imprecise Markov chains has achieved significant progress in recent years. Its applicability, however, is still very much limited, due in large part to the lack of efficient computational methods for calculating higher-dimensional models. The high computational complexity shows itself especially in the calculation of the imprecise version of the Kolmogorov backward equation. The equation is represented at every point of an interval in the form of a minimization problem, solvable merely with linear programming techniques. Consequently, finding an exact solution on an entire interval is infeasible, whence approximation approaches have been developed. To achieve sufficient accuracy, in general, the linear programming optimization methods need to be used in a large number of time points. The principal goal of this paper is to provide a new, more efficient approach for solving the imprecise Kolmogorov backward equation. It is based on the Lipschitz continuity of the solutions of the equation with respect to time, causing the linear programming problems appearing in proximate points of the time interval to have similar optimal solutions. This property is exploited by utilizing the theory of normal cones of convex sets. The present article is primarily devoted to providing the theoretical basis for the novel technique, yet, the initial testing shows that in most cases it decisively outperforms the existing methods.