Concise Reasoning in the Lens of Lagrangian Optimization

Concise reasoning in large language models seeks to generate only essential intermediate steps needed to arrive at a final answer, thereby alleviating issues of "over-thinking". Most proposed approaches hinge on carefully hand-crafted heuristics, struggling to balance concision with performance, often failing to adapt across domains and model scales. In this work, we address these challenges by introducing a principled and pragmatic strategy, performance-aware length updating (P ALU). As a principled algorithm, P ALU formulates concise reasoning as a constrained optimization problem, minimizing response length subject to a performance constraint, and then applies Lagrangian optimization to convert it into a tractable unconstrained problem. As a pragmatic solution, P ALU streamlines complicated update rules through three approximations: (i) estimating performance with off-policy rollouts, (ii) truncating the Lagrange multiplier to two extremes, and (iii) replacing gradient-based updates with quantile-driven length adjustments. Furthermore, P ALU is demonstrated to adapt across both domain (logic, STEM and math) and model scale (1.5B, 7B, 14B) entrenching the algorithm as a practical and effective concise reasoning approach. Reasoning, requiring large language models (LLMs) to work through intermediate steps before producing a final answer, substantially improves performance on complex tasks such as mathematics (Jaech et al., 2024; Shao et al., 2024), programming (Lambert et al., 2024), and value alignment (Guo et al., 2025). Y et this benefit is often accompanied by overthinking: redundant self-reflection, backtracking, and validation (Chen et al., 2024; Zhang et al., 2024; Fatemi et al., 2025). These limitations inflate inference costs and hampers user experience, motivating the need for concise reasoning--the production of only the essential steps required to reach a correct answer.