Robust Shortest Path Planning and Semicontractive Dynamic Programming

TL;DR

This paper addresses robust shortest path problems under uncertainty using semicontractive dynamic programming, proposing algorithms with theoretical guarantees for existence, uniqueness, and optimality of solutions.

Contribution

It introduces a novel framework combining robust shortest path planning with semicontractive dynamic programming, extending classical algorithms to uncertain environments.

Findings

Existence and uniqueness of solutions established.

Algorithms for value and policy iteration adapted for robustness.

Dijkstra-like algorithm proven effective for nonnegative costs.

Abstract

In this paper we consider shortest path problems in a directed graph where the transitions between nodes are subject to uncertainty. We use a minimax formulation, where the objective is to guarantee that a special destination state is reached with a minimum cost path under the worst possible instance of the uncertainty. Problems of this type arise, among others, in planning and pursuit-evasion contexts, and in model predictive control. Our analysis makes use of the recently developed theory of abstract semicontractive dynamic programming models. We investigate questions of existence and uniqueness of solution of the optimality equation, existence of optimal paths, and the validity of various algorithms patterned after the classical methods of value and policy iteration, as well as a Dijkstra-like algorithm for problems with nonnegative arc lengths.