Some advances on constrained Markov decision processes in Borel spaces with random state-dependent discount factors

ABSTRACT

This paper addresses a class of discrete-time Markov decision processes in Borel spaces with a finite number of cost constraints. The constrained control model considers costs of discounted type with state-dependent discount factors which are subject to external disturbances. Our objective is to prove the existence of optimal control policies and characterize them according to certain optimality criteria. Specifically, by rewriting appropriately our original constrained problem as a new one on a space of occupation measures, we apply the direct method to show solvability. Next, the problem is defined as a convex program, and we prove that the existence of a saddle point of the associated Lagrangian operator is equivalent to the existence of an optimal control policy for the constrained problem. Finally, we turn our attention to multi-objective optimization problems, where the existence of Pareto optimal policies can be obtained from the existence of saddle-points of the aforementioned Lagrangian or equivalently from the existence of optimal control policies of constrained problems.

KEYWORDS:

• Markov decision processes
• random non-constant discount factor
• constrained control problems
• convex programming
• Pareto optimality

AMS 2020 SUBJECT CLASSIFICATIONS:

• 93E20
• 90C40

Disclosure statement

The authors declare that they have no conflict of interest.

Additional information

Funding

Work partially supported by Consejo Nacional de Ciencia y Tecnología (CONACYT) – México [grant number Ciencia Frontera 2019-87787] and [grant number PRODEP-2021 no. CA-38].