Idempotent structures in optimization (Q2777856)

This paper deals with application of the idempotent analysis to different problems of optimization. The author gives some introduction to the idempotent analysis. Notions of idempotent metric semi-ring (semi-group) and some examples of idempotent semi-rings are presented. Properties of the idempotent measures, idempotent integrals, linear operators in idempotent semi-module are described. Applications of the idempotent spectral analysis to the investigation of dynamic optimization problems with planned infinite-horizon and to models of mathematical economics are presented. Then the author introduces elements of non-linear idempotent analysis and shows that the theory of additive homogeneous operators has a deep connection with game theory. Using these results the turnpike theorem for stochastic games is obtained. The idempotent structures help to define and construct generalized solutions of the Hamilton-Jacobi-Bellman equation and to study the behavior of these solutions. The perturbation theory of solutions of the deterministic Bellman equation with random noise are studied. Applications of the idempotent analysis to multicriteria optimization, to stochastic and infinite-dimensional Hamilton-Jacobi-Bellman equations, and to problems of financial mathematics are presented.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00043].