A More Scalable Mixed-Integer Encoding for Metric Temporal Logic

Abstract: The state-of-the-art in optimal control from timed temporal logic specifications, including Metric Temporal Logic (MTL) and Signal Temporal Logic (STL), is based on Mixed-Integer Convex Programming (MICP). The standard MICP approach is sound and complete, but struggles to scale to long and complex specifications. Drawing on recent advances in trajectory optimization for piecewise-affine systems, we propose a new MICP encoding for finite transition systems that significantly improves scalability to long and complex MTL specifications. Rather than seeking to reduce the number of variables in the MICP, we focus instead on designing an encoding with a tight convex relaxation. This leads to a larger optimization problem, but significantly improves branch-and-bound solver performance. In simulation experiments involving a mobile robot in a grid-world, the proposed encoding can reduce computation times by several orders of magnitude.