Multi-Agent Temporal Logic Planning via Penalty Functions and Block-Coordinate Optimization

Abstract

Multi-agent planning under Signal Temporal Logic (STL) is often hindered by collaborative tasks that lead to computational challenges due to the inherent high-dimensionality of the problem, preventing scalable synthesis with satisfaction guarantees. To address this, we formulate STL planning as an optimization program under arbitrary multi-agent constraints and introduce a penalty-based unconstrained relaxation that can be efficiently solved via a Block-Coordinate Gradient Descent (BCGD) method, where each block corresponds to a single agent's decision variables, thereby mitigating complexity. By utilizing a quadratic penalty function defined via smooth STL semantics, we show that BCGD iterations converge to a stationary point of the penalized problem under standard regularity assumptions. To enforce feasibility, the BCGD solver is embedded within a two-layer optimization scheme: inner BCGD updates are performed for a fixed penalty parameter, which is then increased in an outer loop to progressively improve multi-agent STL robustness. The proposed framework enables scalable computations and is validated through various complex multi-robot planning scenarios.