A key challenge in autonomous driving is that Autonomous Vehicles (AVs) must contend with multiple, often conflicting, planning requirements. These requirements naturally form in a hierarchy -- e.g., avoiding a collision is more important than maintaining lane. While the exact structure of this hierarchy remains unknown, to progress towards ensuring that AVs satisfy pre-determined behavior specifications, it is crucial to develop approaches that systematically account for it. Motivated by lexicographic behavior specification in AVs, this work addresses a lexicographic multi-objective motion planning problem, where each objective is incomparably more important than the next -- consider that avoiding a collision is incomparably more important than a lane change violation. This work ties together two elements. Firstly, a multi-objective candidate function that asymptotically represents lexicographic orders is introduced. Unlike existing multi-objective cost function formulations, this approach assures that returned solutions asymptotically align with the lexicographic behavior specification. Secondly, inspired by continuation methods, we propose two algorithms that asymptotically approach minimum rank decisions -- i.e., decisions that satisfy the highest number of important rules possible. Through a couple practical examples, we showcase that the proposed candidate function asymptotically represents the lexicographic hierarchy, and that both proposed algorithms return minimum rank decisions, even when other approaches do not.
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