An Iterative Scheme for the Approximate Linear Programming Solution to the Optimal Control of a Markov Decision Process

This paper addresses the computational issues involved in the solution to an infinite-horizon optimal control problem for a Markov Decision Process (MDP) with a continuous state component and a discrete control input. The optimal Markov policy for the MDP can be determined based on the fixed point solution to the Bellman equation, which can be rephrased as a constrained Linear Program (LP) with an infinite number of constraints and an infinite dimensional optimization variable (the optimal value function). To compute an (approximate) solution to the LP, an iterative randomized scheme is proposed where the optimization variable is expressed as a linear combination of basis functions in a given class: at each iteration, the resulting semi-infinite LP is solved via constraint sampling, whereas the number of basis functions is progressively increased through the iterations so as to meet some performance goal. The effectiveness of the proposed scheme is shown on a multi-room heating system example

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

From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming

We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of randomized optimization and first order methods, leading to a priori as well as a posterior performance guarantees. We illustrate the generality and implications of our theoretical results in the special case of the long-run average cost and discounted cost optimal control problems for Markov decision processes on Borel spaces. The applicability of the theoretical results is demonstrated through a constrained linear quadratic optimal control problem and a fisheries management problem.Comment: 30 pages, 5 figure

Delay-Optimal User Scheduling and Inter-Cell Interference Management in Cellular Network via Distributive Stochastic Learning

In this paper, we propose a distributive queueaware intra-cell user scheduling and inter-cell interference (ICI) management control design for a delay-optimal celluar downlink system with M base stations (BSs), and K users in each cell. Each BS has K downlink queues for K users respectively with heterogeneous arrivals and delay requirements. The ICI management control is adaptive to joint queue state information (QSI) over a slow time scale, while the user scheduling control is adaptive to both the joint QSI and the joint channel state information (CSI) over a faster time scale. We show that the problem can be modeled as an infinite horizon average cost Partially Observed Markov Decision Problem (POMDP), which is NP-hard in general. By exploiting the special structure of the problem, we shall derive an equivalent Bellman equation to solve the POMDP problem. To address the distributive requirement and the issue of dimensionality and computation complexity, we derive a distributive online stochastic learning algorithm, which only requires local QSI and local CSI at each of the M BSs. We show that the proposed learning algorithm converges almost surely (with probability 1) and has significant gain compared with various baselines. The proposed solution only has linear complexity order O(MK)