The Linear Programming Approach to Reach-Avoid Problems for Markov Decision Processes

TL;DR

This paper presents a linear programming framework for solving reach-avoid problems in Markov decision processes, providing a tractable approximation method that is validated through numerical case studies.

Contribution

It introduces a novel linear programming approach for stochastic reach-avoid problems and develops finite dimensional approximations for efficient computation.

Findings

The method effectively computes control policies for reach-avoid specifications.

Finite dimensional approximations reduce computational complexity.

Numerical case studies demonstrate the approach's practical viability.

Abstract

One of the most fundamental problems in Markov decision processes is analysis and control synthesis for safety and reachability specifications. We consider the stochastic reach-avoid problem, in which the objective is to synthesize a control policy to maximize the probability of reaching a target set at a given time, while staying in a safe set at all prior times. We characterize the solution to this problem through an infinite dimensional linear program. We then develop a tractable approximation to the infinite dimensional linear program through finite dimensional approximations of the decision space and constraints. For a large class of Markov decision processes modeled by Gaussian mixtures kernels we show that through a proper selection of the finite dimensional space, one can further reduce the computational complexity of the resulting linear program. We validate the proposed method and analyze its potential with a series of numerical case studies.