Constrained markov decision processes with compact state and action spaces: the average case

Constrained Markov decision processes with compact state and action spaces are studied under long-run average reward or cost criteria. By introducing a corresponding Lagrange function, a saddle-point theorem is given, by which the existence of a constrained optimal pair of initial state distribution and policy is shown. Also, under the hypothesis of Doeblin, the functional characterization of a constrained optimal policy is obtained     

Average optimality for continuous-time Markov decision processes with a policy iteration approach

This paper deals with the average expected reward criterion for continuous-time Markov decision processes in general state and action spaces. The transition rates of underlying continuous-time jump Markov processes are allowed to be unbounded, and the reward rates may have neither upper nor lower bounds. We give conditions on the system's primitive data and under which we prove the existence of the average reward optimality equation and an average optimal stationary policy. Also, under our conditions we ensure the existence of ?-average optimal stationary policies. Moreover, we study some properties of average optimal stationary policies. We not only establish another average optimality equation on an average optimal stationary policy, but also present an interesting “martingale characterization” of such a policy. The approach provided in this paper is based on the policy iteration algorithm. It should be noted that our way is rather different from both the usually “vanishing discounting factor approach” and the “optimality inequality approach” widely used in the previous literature.     

New Average Optimality Conditions for Semi-Markov Decision Processes in Borel Spaces

This paper deals with semi-Markov decision processes under the average expected criterion. The state and action spaces are Borel spaces, and the cost/reward function is allowed to be unbounded from above and from below. We give another set of conditions, under which the existence of an optimal (deterministic) stationary policy is proven by a new technique of two average optimality inequalities. Our conditions are slightly weaker than those in the existing literature, and some new sufficient conditions for the verifications of our assumptions are imposed on the primitive data of the model. Finally, we illustrate our results with three examples.     

Average optimality inequality for continuous-time Markov decision processes in Polish spaces

In this paper, we study the average optimality for continuous-time controlled jump Markov processes in general state and action spaces. The criterion to be minimized is the average expected costs. Both the transition rates and the cost rates are allowed to be unbounded. We propose another set of conditions under which we first establish one average optimality inequality by using the well-known “vanishing discounting factor approach”. Then, when the cost (or reward) rates are nonnegative (or nonpositive), from the average optimality inequality we prove the existence of an average optimal stationary policy in all randomized history dependent policies by using the Dynkin formula and the Tauberian theorem. Finally, when the cost (or reward) rates have neither upper nor lower bounds, we also prove the existence of an average optimal policy in all (deterministic) stationary policies by constructing a “new” cost (or reward) rate. Research partially supported by the Natural Science Foundation of China (Grant No: 10626021) and the Natural Science Foundation of Guangdong Province (Grant No: 06300957).     

非一致有界费用MDP的强平均最优性条件

研究可数状态空间任意行动空间非一致性有界费用马氏决策过程(MDP)的强平均最优,给出了使得每个常用的平均最优策略也是强平均最优的条件,并实质性的推广了Cavazos-Cadena和Fernandez-Gaucheran(Math. Meth. Oper. Res., 1996, 43: 281-300)的主要结果.     

Sample path optimality for a Markov optimization problem

We study a unichain Markov decision process i.e. a controlled Markov process whose state process under a stationary policy is an ergodic Markov chain. Here the state and action spaces are assumed to be either finite or countable. When the state process is uniformly ergodic and the immediate cost is bounded then a policy that minimizes the long-term expected average cost also has an nth stage sample path cost that with probability one is asymptotically less than the nth stage sample path cost under any other non-optimal stationary policy with a larger expected average cost. This is a strengthening in the Markov model case of the a.s. asymptotically optimal property frequently discussed in the literature.     

Average cost Markov decision processes under the hypothesis of Doeblin

Average cost Markov decision processes (MDPs) with compact state and action spaces and bounded lower semicontinuous cost functions are considered. Kurano [7] has treated the general case in which several ergodic classes and a transient set are permitted for the Markov process induced by any randomized stationary policy under the hypothesis of Doeblin and showed the existence of a minimum pair of state and policy. This paper considers the same case as that discussed in Kurano [7] and proves some new results which give the existence theorem of an optimal stationary policy under some reasonable conditions.     

The Discounted Method and Equivalence of Average Criteria for Risk-Sensitive Markov Decision Processes on Borel Spaces

This note concerns discrete-time controlled Markov chains with Borel state and action spaces. Given a nonnegative cost function, the performance of a control policy is measured by the superior limit risk-sensitive average criterion associated with a constant and positive risk sensitivity coefficient. Within such a framework, the discounted approach is used (a) to establish the existence of solutions for the corresponding optimality inequality, and (b) to show that, under mild conditions on the cost function, the optimal value functions corresponding to the superior and inferior limit average criteria coincide on a certain subset of the state space. The approach of the paper relies on standard dynamic programming ideas and on a simple analytical derivation of a Tauberian relation.     

Constrained semi-Markov decision processes with ratio and time expected average criteria in Polish spaces

This paper deals with the ratio and time expected average criteria for constrained semi-Markov decision processes (SMDPs). The state and action spaces are Polish spaces, the rewards and costs are unbounded from above and from below, and the mean holding times are allowed to be unbounded from above. First, under general conditions we prove the existence of constrained-optimal policies for the ratio expected average criterion by developing a technique of occupation measures including the mean holding times for SMDPs, which are the generalizations of those for the standard discrete-time and continuous-time MDPs. Then, we give suitable conditions under which we establish the equivalence of the two average criteria by the optional sampling theorem, and thus we show the existence of constrained-optimal policies for the time expected average criterion. Finally, we illustrate the application of our main results with a controlled linear system, for which an exact optimal policy is obtained.     

Stochastic Games with Average Payoff Criterion

We study two-person stochastic games on a Polish state and compact action spaces and with average payoff criterion under a certain ergodicity condition. For the zero-sum game we establish the existence of a value and stationary optimal strategies for both players. For the nonzero-sum case the existence of Nash equilibrium in stationary strategies is established under certain separability conditions. Accepted 9 January 1997     

Nonstationary denumerable state Markov decision processes – with average variance criterion

In this paper, we consider the nonstationary Markov decision processes (MDP, for short) with average variance criterion on a countable state space, finite action spaces and bounded one-step rewards. From the optimality equations which are provided in this paper, we translate the average variance criterion into a new average expected cost criterion. Then we prove that there exists a Markov policy, which is optimal in an original average expected reward criterion, that minimizies the average variance in the class of optimal policies for the original average expected reward criterion.     

Existence of optimal policy for time non-homogeneous discounted Markovian decision programming

In this paper we discuss the discrete time non-homogeneous discounted Markovian decision programming, where the state space and all action sets are countable. Suppose that the optimum value function is finite. We give the necessary and sufficient conditions for the existence of an optimal policy. Suppose that the absolute mean of rewards is relatively bounded. We also give the necessary and sufficient conditions for the existence of an optimal policy.     

First passage Markov decision processes with constraints and varying discount factors

This paper focuses on the constrained optimality problem (COP) of first passage discrete-time Markov decision processes (DTMDPs) in denumerable state and compact Borel action spaces with multi-constraints, state-dependent discount factors, and possibly unbounded costs. By means of the properties of a so-called occupation measure of a policy, we show that the constrained optimality problem is equivalent to an (infinite-dimensional) linear programming on the set of occupation measures with some constraints, and thus prove the existence of an optimal policy under suitable conditions. Furthermore, using the equivalence between the constrained optimality problem and the linear programming, we obtain an exact form of an optimal policy for the case of finite states and actions. Finally, as an example, a controlled queueing system is given to illustrate our results.     

Equivalence of Lyapunov stability criteria in a class of Markov decision processes

We are concerned with Markov decision processes with countable state space and discrete-time parameter. The main structural restriction on the model is the following: under the action of any stationary policy the state space is acommunicating class. In this context, we prove the equivalence of ten stability/ergodicity conditions on the transition law of the model, which imply the existence of average optimal stationary policies for an arbitrary continuous and bounded reward function; these conditions include the Lyapunov function condition (LFC) introduced by A. Hordijk. As a consequence of our results, the LFC is proved to be equivalent to the following: under the action of any stationary policy the corresponding Markov chain has a unique invariant distribution which depends continuously on the stationary policy being used. A weak form of the latter condition was used by one of the authors to establish the existence of optimal stationary policies using an approach based on renewal theory.This research was supported in part by the Third World Academy of Sciences (TWAS) under Grant TWAS RG MP 898-152.     

Discounted continuous-time Markov decision processes with unbounded rates and randomized history-dependent policies: the dynamic programming approach

This paper deals with a continuous-time Markov decision process in Borel state and action spaces and with unbounded transition rates. Under history-dependent policies, the controlled process may not be Markov. The main contribution is that for such non-Markov processes we establish the Dynkin formula, which plays important roles in establishing optimality results for continuous-time Markov decision processes. We further illustrate this by showing, for a discounted continuous-time Markov decision process, the existence of a deterministic stationary optimal policy (out of the class of history-dependent policies) and characterizing the value function through the Bellman equation.     

Approximation of Markov decision processes with general state space

We deal with a discrete-time finite horizon Markov decision process with locally compact Borel state and action spaces, and possibly unbounded cost function. Based on Lipschitz continuity of the elements of the control model, we propose a state and action discretization procedure for approximating the optimal value function and an optimal policy of the original control model. We provide explicit bounds on the approximation errors. Our results are illustrated by a numerical application to a fisheries management problem.     

Solutions of the average cost optimality equation for finite Markov decision chains: risk-sensitive and risk-neutral criteria

This work is concerned with controlled Markov chains with finite state and action spaces. It is assumed that the decision maker has an arbitrary but constant risk sensitivity coefficient, and that the performance of a control policy is measured by the long-run average cost criterion. Within this framework, the existence of solutions of the corresponding risk-sensitive optimality equation for arbitrary cost function is characterized in terms of communication properties of the transition law.     

Optimal policies for constrained average-cost Markov decision processes

We give mild conditions for the existence of optimal solutions for a Markov decision problem with average cost, under m constraints of the same kind, in Borel actions and states spaces. Moreover, there is an optimal policy that is a convex combination of at most m+1 deterministic policies.     

A mean–variance optimization problem for discounted Markov decision processes

In this paper, we consider a mean–variance optimization problem for Markov decision processes (MDPs) over the set of (deterministic stationary) policies. Different from the usual formulation in MDPs, we aim to obtain the mean–variance optimal policy that minimizes the variance over a set of all policies with a given expected reward. For continuous-time MDPs with the discounted criterion and finite-state and action spaces, we prove that the mean–variance optimization problem can be transformed to an equivalent discounted optimization problem using the conditional expectation and Markov properties. Then, we show that a mean–variance optimal policy and the efficient frontier can be obtained by policy iteration methods with a finite number of iterations. We also address related issues such as a mutual fund theorem and illustrate our results with an example.