Application of Markov decision processes to search problems

Many decision problems contain, in some form, a NP-hard combinatorial problem. Therefore decision support systems have to solve such combinatorial problems in a reasonable time. Many combinatorial problems can be solved by a search method. The search methods used in decision support systems have to be robust in the sense that they can handle a large variety of (user defined) constraints and that they allow user interaction, i.e. they allow a decision maker to control the search process manually. In this paper we show how Markov decision processes can be used to guide a random search process. We first formulate search problems as a special class of Markov decision processes such that the search space of a search problem is the state space of the Markov decision process. In general it is not possible to compute an optimal control procedure for these Markov decision processes in a reasonable time. We therefore, define several simplifications of the original problem that have much smaller state spaces. For these simplifications, decompositions and abstractions, we find optimal strategies and use the exact solutions of these simplified problems to guide a randomized search process. The search process selects states for further search at random with probabilities based on the optimal strategies of the simplified problems. This randomization is a substitute for explicit backtracking and avoids problems with local extrema. These randomized search procedures are repeated as long as we have time to solve the problem. The best solution of those generated during that time is accepted. We illustrate the approach with two examples: the N-puzzle and a job shop scheduling problem

IST Austria Technical Report

We consider partially observable Markov decision processes (POMDPs) with ω-regular conditions specified as parity objectives. The class of ω-regular languages extends regular languages to infinite strings and provides a robust specification language to express all properties used in verification, and parity objectives are canonical forms to express ω-regular conditions. The qualitative analysis problem given a POMDP and a parity objective asks whether there is a strategy to ensure that the objective is satis- fied with probability 1 (resp. positive probability). While the qualitative analysis problems are known to be undecidable even for very special cases of parity objectives, we establish decidability (with optimal complexity) of the qualitative analysis problems for POMDPs with all parity objectives under finite- memory strategies. We establish asymptotically optimal (exponential) memory bounds and EXPTIME- completeness of the qualitative analysis problems under finite-memory strategies for POMDPs with parity objectives

Multi-objective Robust Strategy Synthesis for Interval Markov Decision Processes

Interval Markov decision processes (IMDPs) generalise classical MDPs by having interval-valued transition probabilities. They provide a powerful modelling tool for probabilistic systems with an additional variation or uncertainty that prevents the knowledge of the exact transition probabilities. In this paper, we consider the problem of multi-objective robust strategy synthesis for interval MDPs, where the aim is to find a robust strategy that guarantees the satisfaction of multiple properties at the same time in face of the transition probability uncertainty. We first show that this problem is PSPACE-hard. Then, we provide a value iteration-based decision algorithm to approximate the Pareto set of achievable points. We finally demonstrate the practical effectiveness of our proposed approaches by applying them on several case studies using a prototypical tool.Comment: This article is a full version of a paper accepted to the Conference on Quantitative Evaluation of SysTems (QEST) 201

Some numerical methods for solving stochastic impulse control in natural gas storage facilities

The valuation of gas storage facilities is characterized as a stochastic impulse control problem with finite horizon resulting in Hamilton-Jacobi-Bellman (HJB) equations for the value function. In this context the two catagories of solving schemes for optimal switching are discussed in a stochastic control framework. We reviewed some numerical methods which include approaches related to partial differential equations (PDEs), Markov chain approximation, nonparametric regression, quantization method and some practitioners’ methods. This paper considers optimal switching problem arising in valuation of gas storage contracts for leasing the storage facilities, and investigates the recent developments as well as their advantages and disadvantages of each scheme based on dynamic programming principle (DPP