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This brief paper presents simple simulation-based algorithms for obtaining an approximately optimal policy in a given finite set in large finite constrained Markov decision processes. The algorithms are adapted from playing strategies for "sleeping experts and bandits" problem and their computational complexities are independent of state and action space sizes if the given policy set is relatively small. We establish convergence of their expected performances to the value of an optimal policy and convergence rates, and also almost-sure convergence to an optimal policy with an exponential rate for the algorithm adapted within the context of sleeping experts.

This paper, based on the compactness-continuity and finite value conditions, establishes the sufficiency of the class of stationary policies out of the general class of history-dependent ones for a constrained continuous-time Markov decision process in Borel state and action spaces with total nonnegative cost criteria. The controlled process is not necessarily absorbing, the discount factor can be identically equal to zero, and the transition rates can also be equal to zero, which account for the major technical difficulties. Models with these features seemingly had not been handled in the previous literature.

This paper is devoted to studying constrained continuous-time Markov decision processes (MDPs) in the class of randomized policies depending on state histories. The transition rates may be unbounded, the reward and costs are admitted to be unbounded from above and from below, and the state and action spaces are Polish spaces. The optimality criterion to be maximized is the expected discounted rewards, and the constraints can be imposed on the expected discounted costs. First, we give conditions for the nonexplosion of underlying processes and the finiteness of the expected discounted rewards/costs. Second, using a technique of occupation measures, we prove that the constrained optimality of continuous-time MDPs can be transformed to an equivalent (optimality) problem over a class of probability measures. Based on the equivalent problem and a so-called $\bar{w}$-weak convergence of probability measures developed in this paper, we show the existence of a constrained optimal policy. Third, by providing a linear programming formulation of the equivalent problem, we show the solvability of constrained optimal policies. Finally, we use two computable examples to illustrate our main results.

This paper is devoted to studying constrained continuous-time Markov decision processes (MDPs) in the class of randomized policies depending on state histories. The transition rates may be unbounded, the reward and costs are admitted to be unbounded from above and from below, and the state and action spaces are Polish spaces. The optimality criterion to be maximized is the expected discounted rewards, and the constraints can be imposed on the expected discounted costs. First, we give conditions for the nonexplosion of underlying processes and the finiteness of the expected discounted rewards/costs. Second, using a technique of occupation measures, we prove that the constrained optimality of continuous-time MDPs can be transformed to an equivalent (optimality) problem over a class of probability measures. Based on the equivalent problem and a so-called $\bar{w}$-weak convergence of probability measures developed in this paper, we show the existence of a constrained optimal policy. Third, by providing a linear programming formulation of the equivalent problem, we show the solvability of constrained optimal policies. Finally, we use two computable examples to illustrate our main results.

We present stochastic approximation algorithms for computing the locally optimal policy of a constrained, average cost, finite state Markov Decision Process. The stochastic approximation algorithms require estimation of the gradient of the cost function with respect to the parameter that characterizes the randomized policy. We propose a spherical coordinate parameterization and implement a simulation based gradient estimation scheme using using potential realization factors. In this paper we analyse boundedness of moments of the estimators and provide a new analysis for the computational requirements of the estimators. We present a stochastic version of a primal dual (augmented) Lagrange multiplier method for the constrained algorithm. We show that the "large sample size" limit of the procedure is unbiased, but for small sample size (which is relevant for on-line real time operation) it is biased. We give an explicit expression of the asymptotic bias and discuss several possibilities for bias reduction. Numerical examples are given to illustrate the performance of the algorithms. In particular we present a case study in transmission control in wireless communications where the optimal policy has a simplified threshold structure and apply our stochastic approximation method to this example.