Geometric Pareto Control: Riemannian Gradient Flow of Energy Function via Lie Group Homotopy

TL;DR

The paper introduces Geometric Pareto Control, a Riemannian gradient flow framework on Lie groups that enables safe, continuous, and retraining-free multi-objective control in cyber-physical systems.

Contribution

It develops a geometric two-stage approach embedding Pareto solutions in a Lie group, facilitating online control without retraining and ensuring feasibility under system uncertainties.

Findings

Achieves 100% feasibility in control tasks and power flow.

Maintains continuous control actions under changing conditions.

Outperforms model-free baselines in feasibility and adaptability.

Abstract

We propose Geometric Pareto Control (GPC), a framework overcoming barriers of reinforcement learning in cyber-physical systems where governing physics is known. Reinforcement learning confronts barriers in safety-critical applications: sample complexity grows with action-space dimension, retraining is required when objectives or conditions shift, goals such as safety recovery and economic dispatch demand brittle switching logic, and unsafe exploration persists under constrained RL formulations. GPC resolves these barriers through a two-stage geometric approach. Offline, the supported family of Pareto-optimal solutions (i.e., solutions recoverable by weighted scalarization) is embedded as a submanifold within a Lie group. Exponential map closure preserves membership in the ambient Lie group; drift and reset assumptions keep online latent states within a bounded neighbourhood of the Pareto submanifold, and a training-time feasibility margin guarantees decoded actions remain feasible without post-hoc projection, constructing a "map" of the solution landscape. Online, a closed-form proximal navigator traverses this submanifold via a unified Riemannian gradient flow driven by a singular perturbation potential field, inducing dual-timescale dynamics that prioritize constraint restoration over performance optimization. The homeomorphic structure of the submanifold guarantees that varying system parameters and objective weights produce continuous control actions, enabling deployment under unseen conditions without retraining. Validated on a nonconvex control task and real-time multi-objective optimal power flow, GPC achieves 100% feasibility, 0.30% oracle suboptimality, and 12.3 ms decisions while shifting from constraint recovery to economic dispatch. Under branch-admittance uncertainty, it remains 100% feasible without retraining, whereas model-free baselines produce no feasible dispatches.