Multi-Objective LQR with Linear Scalarization

TL;DR

This paper characterizes the Pareto front in multi-objective LQR as linear scalarization, demonstrating that tradeoffs can be efficiently approximated and extended to uncertain dynamics, advancing multi-objective control methods.

Contribution

It proves that linear scalarization fully characterizes the Pareto front in multi-objective LQR and introduces an efficient approximation algorithm, extending analysis to uncertain dynamics.

Findings

Pareto front in multi-objective LQR is linear scalarizable.

A grid search over scalarization parameters approximates the Pareto front.

The approach extends to uncertain dynamics with certainty equivalence.

Abstract

The framework of decision-making, modeled as a Markov Decision Process (MDP), typically assumes a single objective. However, practical scenarios often involve tradeoffs between multiple objectives. We address this in the Linear Quadratic Regulator (LQR), a canonical continuous, infinite horizon MDP. First, we establish that the Pareto front for LQR is characterized by linear scalarization: a convex combination of objectives recovers all tradeoff points, making multi-objective LQR reducible to single-objective problems. This highlights an important instance where linear scalarization suffices for a non-convex problem. Second, we show the Pareto front is smooth, in that an $\epsilon$ perturbation of a scalarization parameter yields an $\epsilon$ approximation to the objective. These results inspire a simple algorithm to approximate the Pareto front via grid search over scalarization parameters, where each optimization problem retains the computational efficiency of single-objective LQR. Lastly, we extend the analysis to certainty equivalence, where unknown dynamics are replaced with estimates.