In this paper, we consider risk-sensitive discounted control problem for continuous-time jump Markov processes taking values in general state space. The transition rates of underlying continuous-time jump Markov processes and the cost rates are allowed to be unbounded. Under certain Lyapunov condition, we establish the existence and uniqueness of the solution to the Hamilton-Jacobi-Bellman (HJB) equation. Also we prove the existence of optimal risk-sensitive control in the class of Markov control.
In this article we consider risk-sensitive control of semi-Markov processes with a discrete state space. We consider general utility functions and discounted cost in the optimization criteria. We consider random finite horizon and infinite horizon problems. Using a state augmentation technique we characterise the value functions and also prescribe optimal controls.
In this paper, we investigate a partially observable zero sum games where the state process is a discrete time Markov chain. We consider a general utility function in the optimization criterion. We show the existence of value for both finite and infinite horizon games and also establish the existence of optimal polices. The main step involves converting the partially observable game into a completely observable game which also keeps track of the total discounted accumulated reward/cost.
We study discrete-time discounted constrained Markov decision processes (CMDPs) on Borel spaces with unbounded reward functions. In our approach the transition probability functions are weakly or set-wise continuous. The reward functions are upper semicontinuous in state-action pairs or semicontinuous in actions. Our aim is to study models with unbounded reward functions, which are often encountered in applications, e.g., in consumption/investment problems. We provide some general assumptions under which the optimization problems in CMDPs are solvable in the class of stationary randomized poli…
In this paper, we consider a large class of constrained non-cooperative stochastic Markov games with countable state spaces and discounted cost criteria. In one-player case, i.e., constrained discounted Markov decision models, it is possible to formulate a static optimisation problem whose solution determines a stationary optimal strategy (alias control or policy) in the dynamical infinite horizon model. This solution lies in the compact convex set of all occupation measures induced by strategies, defined on the set of state-action pairs. In case of n-person discounted games the occupation mea…
This note re-visits the rolling-horizon control approach to the problem of a Markov decision process (MDP) with infinite-horizon discounted expected reward criterion. Distinguished from the classical value-iteration approach, we develop an asynchronous on-line algorithm based on policy iteration integrated with a multi-policy improvement method of policy switching. A sequence of monotonically improving solutions to the forecast-horizon sub-MDP is generated by updating the current solution only at the currently visited state, building in effect a rolling-horizon control policy for the MDP over …
This paper studies a continuous-review backlogged inventory model considered by Helmes et al. (2015) but with discontinuous quantity-dependent setup cost for each order. In particular, the setup cost is characterized by a two-step function and a higher cost would be charged once the order quantity exceeds a threshold $Q$. Unlike the optimality of $(s,S)$-type policy obtained by Helmes et al. (2015) for continuous setup cost with the discounted cost criterion, we find that, in our model, although some $(s,S)$-type policy is indeed optimal in some cases, the $(s,S)$-type policy can not always be…
In this paper we consider the problem of computing an $ε$-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in $O(1)$ time. Given such a DMDP with states $S$, actions $A$, discount factor $γ\in(0,1)$, and rewards in range $[0, 1]$ we provide an algorithm which computes an $ε$-optimal policy with probability $1 - δ$ where \emph{both} the time spent and number of sample taken are upper bounded by \[ O\left[\frac{|S||A|}{(1-γ)^3 …
We study the synthesis of a policy in a Markov decision process (MDP) following which an agent reaches a target state in the MDP while minimizing its total discounted cost. The problem combines a reachability criterion with a discounted cost criterion and naturally expresses the completion of a task with probabilistic guarantees and optimal transient performance. We first establish that an optimal policy for the considered formulation may not exist but that there always exists a near-optimal stationary policy. We additionally provide a necessary and sufficient condition for the existence of an…
Sun Yat-sen University（中山大学）；Tsinghua University（清华大学）
This paper studies the risk-averse mean-variance optimization in infinite-horizon discounted Markov decision processes (MDPs). The involved variance metric concerns reward variability during the whole process, and future deviations are discounted to their present values. This discounted mean-variance optimization yields a reward function dependent on a discounted mean, and this dependency renders traditional dynamic programming methods inapplicable since it suppresses a crucial property -- time consistency. To deal with this unorthodox problem, we introduce a pseudo mean to transform the untre…
This paper proves continuity of value functions in discounted periodic-review single-commodity total-cost inventory control problems with \revision{continuous inventory levels,} fixed ordering costs, possibly bounded inventory storage capacity, and possibly bounded order sizes for finite and infinite horizons. In each of these constrained models, the finite and infinite-horizon value functions are continuous, there exist deterministic Markov optimal finite-horizon policies, and there exist stationary deterministic Markov optimal infinite-horizon policies. For models with bounded inventory stor…
Fudan University（复旦大学）；Columbia University（哥伦比亚大学）；University of California, Los Angeles（加州大学洛杉矶分校）
In this paper, we propose AsyncQVI, an asynchronous-parallel Q-value iteration for discounted Markov decision processes whose transition and reward can only be sampled through a generative model. Given such a problem with $|\mathcal{S}|$ states, $|\mathcal{A}|$ actions, and a discounted factor $γ\in(0,1)$, AsyncQVI uses memory of size $\mathcal{O}(|\mathcal{S}|)$ and returns an $\varepsilon$-optimal policy with probability at least $1-δ$ using $$\tilde{\mathcal{O}}\big(\frac{|\mathcal{S}||\mathcal{A}|}{(1-γ)^5\varepsilon^2}\log(\frac{1}δ)\big)$$ samples. AsyncQVI is also the first asynchronous…
Recently, Sidford, Wang, Wu and Ye (2018) developed an algorithm combining variance reduction techniques with value iteration to solve discounted Markov decision processes. This algorithm has a sublinear complexity when the discount factor is fixed. Here, we extend this approach to mean-payoff problems, including both Markov decision processes and perfect information zero-sum stochastic games. We obtain sublinear complexity bounds, assuming there is a distinguished state which is accessible from all initial states and for all policies. Our method is based on a reduction from the mean payoff pr…
In this article, we study a discounted stochastic game to model resource optimal intrusion detection in wireless sensor networks. To address the problem of uncertainties in various network parameters, we propose a globalized robust game theoretic framework for discounted robust stochastic games. A robust solution to the considered problem is an optimal point that is feasible for all realizations of data from a given uncertainty set. To allow a controlled violation of the constraints when the parameters move out of the uncertainty set, the concept of globalized robust framework comes into view.…
Optimal Markov Decision Process policies for problems with finite state and action space are identified through a partial ordering by comparing the value function across states. This is referred to as state-based optimality. This paper identifies when such optimality guarantees some form of system-based optimality as measured by a scalar. Four such system-based metrics are introduced. Uni-variate empirical distributions of these metrics are obtained through simulation as to assess whether theoretically optimal policies provide a statistically significant advantage. This has been conducted usin…
We consider a nonzero-sum Markov game on an abstract measurable state space with compact metric action spaces. The goal of each player is to maximize his respective discounted payoff function under the condition that some constraints on a discounted payoff are satisfied. We are interested in the existence of a Nash or noncooperative equilibrium. Under suitable conditions, which include absolute continuity of the transitions with respect to some reference probability measure, additivity of the payoffs and the transition probabilities (ARAT condition), and continuity in action of the payoff func…
This paper attempts to study the optimal stopping time for semi-Markov processes (SMPs) under the discount optimization criteria with unbounded cost rates. In our work, we introduce an explicit construction of the equivalent semi-Markov decision processes (SMDPs). The equivalence is embodied in the value functions of SMPs and SMDPs, that is, every stopping time of SMPs can induce a policy of SMDPs such that the value functions are equal, and vice versa. The existence of the optimal stopping time of SMPs is proved by this equivalence relation. Next, we give the optimality equation of the value …
In this paper, we consider the problem of optimization and learning for constrained and multi-objective Markov decision processes, for both discounted rewards and expected average rewards. We formulate the problems as zero-sum games where one player (the agent) solves a Markov decision problem and its opponent solves a bandit optimization problem, which we here call Markov-Bandit games. We extend Q-learning to solve Markov-Bandit games and show that our new Q-learning algorithms converge to the optimal solutions of the zero-sum Markov-Bandit games, and hence converge to the optimal solutions o…
This work considers two-player zero-sum semi-Markov games with incomplete information on one side and perfect observation. At the beginning, the system selects a game type according to a given probability distribution and informs to Player 1 only. After each stage, the actions chosen are observed by both players before proceeding to the next stage. Firstly, we show the existence of the value function under the expected discount criterion and the optimality equation. Secondly, the existence and iterative algorithm of the optimal policy for Player 1 are introduced through the optimality equation…
In this paper we study a class of risk-sensitive Markovian control problems in discrete time subject to model uncertainty. We consider a risk-sensitive discounted cost criterion with finite time horizon. The used methodology is the one of adaptive robust control combined with machine learning.
This paper deals with the infinite-horizon optimal control problem for Boolean control networks (BCNs) with a discounted-cost criterion. This problem has been investigated in existing studies with algorithms characterized by high computational complexity. We thus attempt to develop more efficient approaches for this problem from a deterministic Markov decision process (DMDP) perspective. First, we show the eligibility of a DMDP to model the control process of a BCN and the existence of an optimal solution. Next, two approaches are developed to handle the optimal control problem in a DMDP. One …
Benign Nonconvex Landscape for Policy Optimization: Infinite-Horizon Discounted MDPs with General State and Action Spaces
Naveen Jindal School of Management, University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金达尔管理学院）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Tech（佐治亚理工学院H.米尔顿·斯图尔特工业与系统工程学院）
This note summarizes the optimization formulations used in the study of Markov decision processes. We consider both the discounted and undiscounted processes under the standard and the entropy-regularized settings. For each setting, we first summarize the primal, dual, and primal-dual problems of the linear programming formulation. We then detail the connections between these problems and other formulations for Markov decision processes such as the Bellman equation and the policy gradient method.
This paper studies average-cost Markov decision processes with semi-uniform Feller transition probabilities. This class of MDPs was recently introduced by the authors to study MDPs with incomplete information. This paper studies the validity of optimality inequalities, the existence of optimal policies, and the approximations of optimal policies by policies optimizing total discounted costs.
The paper deals with stochastic control problems associated with $H_2$ performance indices such as energy or power norms or energy measurements when norms are not defined. They apply to a class of systems for which a stochastic process conveys the underlying uncertainties, known as CSVIU (Control and State Variation Increase Uncertainty). These indices allow various emphases from focusing on the transient behavior with the discounted norm to stricter conditions on stability, steady-state mean-square error and convergence rate, using the optimal overtaking criterion -- the long-run average powe…
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. The main contribution is to show that the problem possesses a solution if it is feasible, and to provide two methods for finding an approximate solution in the form of an ultimately stationary (US) MR policy. The latter is achieve…
