On the Adversarial Convex Body Chasing Problem

TL;DR

This paper extends the convex bodies chasing problem to an adversarial setting, analyzing the strategic interaction between a chasing agent and an adversarial opponent, and providing algorithms with performance guarantees.

Contribution

It introduces an adversarial framework for CBC, proves the continuity of the value function, and develops algorithms for approximating optimal policies with guarantees.

Findings

Performance guarantees for the adversarial CBC problem

Continuity of the optimal value function under certain assumptions

Numerical algorithms for approximating optimal strategies

Abstract

In this work, we extend the convex bodies chasing problem (CBC) to an adversarial setting, where an agent (the Player) is tasked with chasing a sequence of convex bodies generated adversarially by another agent (the Opponent). The Player aims to minimize the total cost associated with its own movements, while the Opponent tries to maximize the same cost. The set of feasible convex bodies is finite and known to both agents, which allows us to provide performance guarantees with max-min optimality. Under certain assumptions, we show the continuity of the optimal value function, and propose an algorithm to numerically approximate the optimal policies for both the Player and the Opponent within a guaranteed tolerance. Finally, the theoretical results are verified through numerical examples.