Approximate solutions to constrained risk-sensitive Markov decision processes

Abstract

This paper considers the problem of finding near-optimal Markovian randomized (MR) policies for finite-state-action, infinite-horizon, constrained risk-sensitive Markov decision processes (CRSMDPs). Constraints are in the form of standard expected discounted cost functions as well as expected risk-sensitive discounted cost functions over finite and infinite horizons. We first show that the aforementioned CRSMDP optimization problem possesses a solution if it is feasible (that is, if there exists a policy which satisfies all the constraints). Secondly, we provide two methods for finding an approximate solution in the form of an ultimately stationary (US) MR policy. The latter is achieved through two approximating finite-horizon CRSMDPs constructed from the original CRSMDP by time-truncating the original objective and constraint cost functions, and suitably perturbing the constraint upper bounds. The first approximation gives a US policy which is ϵ-optimal and feasible for the original problem, while the second approximation gives a near-optimal US policy whose violation of the original constraints is bounded above by a specified tolerance value ϵ. A key step in the proofs is an appropriate choice of a metric that makes the set of infinite-horizon MR policies and the feasible regions of the three CRSMDPs compact, and the objective and constraint functions continuous. We also discuss two applications and use an infinite-horizon risk-sensitive inventory control problem as an example to illustrate how existing solution techniques may be used to solve the two approximate finite-horizon problems mentioned above.