Mapless navigation in unknown and partially observable environments remains challenging for mobile robots, particularly when local minima prevent the robot from making progress toward its goal. Existing local navigation methods often lack an explicit mechanism for escaping such situations, while deep reinforcement learning (DRL) approaches typically learn recovery behaviors implicitly through reward design and policy optimization. In this work, we propose \textbf{LME} (Local-Minimum Escaper), a programmatic hierarchical framework that explicitly generates and reasons subgoals to guide robots out of local-minimum regions. LME operates solely on local observations and selects candidate subgoals using interpretable heuristic criteria that account for both surrounding obstacle geometry and candidate-location safety. A local planner then generates low-level motion commands toward the selected subgoal. This design enables LME to handle environments both with and without local minima within a unified framework, while remaining independent of the underlying local planner and requiring no additional training. Extensive experiments in simulated and real-world environments demonstrate that LME provides robust navigation performance and generalizes to challenging unseen scenarios. Furthermore, the generated subgoals can be used to guide different local planners, substantially improving their ability to escape local minima. Successful deployments on both differential-drive and quadruped robots further demonstrate the practical applicability and generality of the proposed framework.

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