Stochastic approximations of constrained discounted Markov decision processes

We consider a discrete-time constrained Markov decision process under the discounted cost optimality criterion. The state and action spaces are assumed to be Borel spaces, while the cost and constraint functions might be unbounded. We are interested in approximating numerically the optimal discounted constrained cost. To this end, we suppose that the transition kernel of the Markov decision process is absolutely continuous with respect to some probability measure μ  . Then, by solving the linear programming formulation of a constrained control problem related to the empirical probability measure _μn${\mu }_n$ of μ, we obtain the corresponding approximation of the optimal constrained cost. We derive a concentration inequality which gives bounds on the probability that the estimation error is larger than some given constant. This bound is shown to decrease exponentially in n. Our theoretical results are illustrated with a numerical application based on a stochastic version of the Beverton–Holt population model.     

A homotopy approach for infinite horizon discounted Markov decision processes

In this paper we consider a homotopy deformation approach to solving Markov decision process problems by the continuous deformation of a simpler Markov decision process problem until it is identical with the original problem. Algorithms and performance bounds are given.     

Credibilistic Markov decision processes: The average case

Using a concept of random fuzzy variables in credibility theory, we formulate a credibilistic model for unichain Markov decision processes under average criteria. And a credibilistically optimal policy is defined and obtained by solving the corresponding non-linear mathematical programming. Also we give a computational example to illustrate the effectiveness of our new model.     

连续时间Markov决策过程的均值-方差优化问题

本文考虑连续时间Markov决策过程折扣模型的均值-方差优化问题.假设状态空间和行动空间均为Polish空间,转移率和报酬率函数均无界.本文的优化目标是在折扣最优平稳策略类里,选取相应方差最小的策略.本文致力于寻找Polish空间下Markov决策过程均值-方差最优策略存在的条件.利用首次进入分解方法,本文证明均值-方差优化问题可以转化为"等价"的期望折扣优化问题,进而得到关于均值-方差优化问题的"最优方程"和均值-方差最优策略的存在性以及它相应的特征.最后,本文给出若干例子说明折扣最优策略的不唯一性和均值-方差最优策略的存在性.     

Markov ratio decision processes

A finite-state Markov decision process, in which, associated with each action in each state, there are two rewards, is considered. The objective is to optimize the ratio of the two rewards over an infinite horizon. In the discounted version of this decision problem, it is shown that the optimal value is unique and the optimal strategy is pure and stationary; however, they are dependent on the starting state. Also, a finite algorithm for computing the solution is given.     

An envelope theorem and some applications to discounted Markov decision processes

In this paper, an Envelope Theorem (ET) will be established for optimization problems on Euclidean spaces. In general, the Envelope Theorems permit analyzing an optimization problem and giving the solution by means of differentiability techniques. The ET will be presented in two versions. One of them uses concavity assumptions, whereas the other one does not require such kind of assumptions. Thereafter, the ET established will be applied to the Markov Decision Processes (MDPs) on Euclidean spaces, discounted and with infinite horizon. As the first application, several examples (including some economic models) of discounted MDPs for which the et allows to determine the value iteration functions will be presented. This will permit to obtain the corresponding optimal value functions and the optimal policies. As the second application of the ET, it will be proved that under differentiability conditions in the transition law, in the reward function, and the noise of the system, the value function and the optimal policy of the problem are differentiable with respect to the state of the system. Besides, various examples to illustrate these differentiability conditions will be provided. This work was partially supported by Benemérita Universidad Aut ónoma de Puebla (BUAP) under grant VIEP-BUAP 38/EXC/06-G, by Consejo Nacional de Ciencia y Tecnología (CONACYT), and by Evaluation-orientation de la COopération Scientifique (ECOS) under grant CONACyT-ECOS M06-M01.     

An average-value-at-risk criterion for Markov decision processes with unbounded costs

We study the Markov decision processes under the average-valueat-risk criterion. The state space and the action space are Borel spaces, the costs are admitted to be unbounded from above, and the discount factors are state-action dependent. Under suitable conditions, we establish the existence of optimal deterministic stationary policies. Furthermore, we apply our main results to a cash-balance model.     

Markov control processes with randomized discounted cost

In this paper we consider Markov Decision Processes with discounted cost and a random rate in Borel spaces. We establish the dynamic programming algorithm in finite and infinity horizon cases. We provide conditions for the existence of measurable selectors. And we show an example of consumption-investment problem. This research was partially supported by the PROMEP grant 103.5/05/40.     

Finite-horizon piecewise deterministic Markov decision processes with unbounded transition rates

This paper is concerned with the problem of minimizing the expected finite-horizon cost for piecewise deterministic Markov decision processes. The transition rates may be unbounded, and the cost functions are allowed to be unbounded from above and from below. The optimality is over the general history-dependent policies, where the control is continuously acting in time. The infinitesimal approach is employed to establish the associated Hamilton-Jacobi-Bellman equation, via which the existence of optimal policies is proved. An example is provided to verify all the assumptions proposed.     

Risk-sensitive average continuous-time Markov decision processes with unbounded rates

Abstract

In this paper we study the risk-sensitive average cost criterion for continuous-time Markov decision processes in the class of all randomized Markov policies. The state space is a denumerable set, and the cost and transition rates are allowed to be unbounded. Under the suitable conditions, we establish the optimality equation of the auxiliary risk-sensitive first passage optimization problem and obtain the properties of the corresponding optimal value function. Then by a technique of constructing the appropriate approximating sequences of the cost and transition rates and employing the results on the auxiliary optimization problem, we show the existence of a solution to the risk-sensitive average optimality inequality and develop a new approach called the risk-sensitive average optimality inequality approach to prove the existence of an optimal deterministic stationary policy. Furthermore, we give some sufficient conditions for the verification of the simultaneous Doeblin condition, use a controlled birth and death system to illustrate our conditions and provide an example for which the risk-sensitive average optimality strict inequality occurs.     

A useful technique for piecewise deterministic Markov decision processes

This note presents a technique that is useful for the study of piecewise deterministic Markov decision processes (PDMDPs) with general policies and unbounded transition intensities. This technique produces an auxiliary PDMDP from the original one. The auxiliary PDMDP possesses certain desired properties, which may not be possessed by the original PDMDP. We apply this technique to risk-sensitive PDMDPs with total cost criteria, and comment on its connection with the uniformization technique.     

First passage models for denumerable semi-Markov decision processes with nonnegative discounted costs

This paper considers a first passage model for discounted semi-Markov decision processes with denumerable states and nonnegative costs.The criterion to be optimized is the expected discounted cost incurred during a first passage time to a given target set.We first construct a semi-Markov decision process under a given semi-Markov decision kernel and a policy.Then,we prove that the value function satisfies the optimality equation and there exists an optimal(or e-optimal) stationary policy under suitable conditions by using a minimum nonnegative solution approach.Further we give some properties of optimal policies.In addition,a value iteration algorithm for computing the value function and optimal policies is developed and an example is given.Finally,it is showed that our model is an extension of the first passage models for both discrete-time and continuous-time Markov decision processes.     

A new strong optimality criterion for nonstationary Markov decision processes

This paper deals with a new optimality criterion consisting of the usual three average criteria and the canonical triplet (totally so-called strong average-canonical optimality criterion) and introduces the concept of a strong average-canonical policy for nonstationary Markov decision processes, which is an extension of the canonical policies of Herna′ndez-Lerma and Lasserre [16] (pages: 77) for the stationary Markov controlled processes. For the case of possibly non-uniformly bounded rewards and denumerable state space, we first construct, under some conditions, a solution to the optimality equations (OEs), and then prove that the Markov policies obtained from the OEs are not only optimal for the three average criteria but also optimal for all finite horizon criteria with a sequence of additional functions as their terminal rewards (i.e. strong average-canonical optimal). Also, some properties of optimal policies and optimal average value convergence are discussed. Moreover, the error bound in average reward between a rolling horizon policy and a strong average-canonical optimal policy is provided, and then a rolling horizon algorithm for computing strong average ε(>0)-optimal Markov policies is given.