arXiv 2302.084672023-02math.OCmath.PR

动态规划中的最优性原理：一则教学笔记

The Principle of Optimality in Dynamic Programming: A Pedagogical Note

National University of Singapore（新加坡国立大学）

Bar Light

备注：: Corrects the published version's definition of the set of measurable selections to allow approximate maximizers as already used in the proof. The conclusions are unchanged

阅读 PDF

arXiv 页面 ↗HTML 原文 ↗

AI总结：

针对动态规划最优性原理教学中常规避一般状态空间测度论问题、无法覆盖连续状态空间经典模型的不足，本文给出贴现动态规划最优性原理成立的条件与自含简洁证明，并结合文献案例阐释其适用性。

摘要：

最优性原理是动态规划的核心基础，其指出动态优化问题的最优解可通过组合其子问题的最优解得到。尽管该原理具有普适性，但为了规避测度论层面的复杂性，教学中通常仅针对有限或可数状态空间的问题讲解该原理。因此，它无法应用于库存管理、动态定价等具有连续状态空间的经典模型，学生也可能意识不到研究一般状态空间动态规划模型时可能面临的挑战。为解决这一问题，我们给出了贴现动态规划最优性原理成立的条件，并提供了自包含的简洁证明。这些条件清晰揭示了一般状态空间情形下可能出现的难点。我们还列举了文献中的相关案例，包括相对复杂的普遍可测动态规划情形，以及可应用我们的主要结论证明最优性原理成立的有限动态规划简单情形。

英文摘要：

The principle of optimality is a fundamental aspect of dynamic programming, which states that the optimal solution to a dynamic optimization problem can be found by combining the optimal solutions to its sub-problems. While this principle is generally applicable, it is often only taught for problems with finite or countable state spaces in order to sidestep measure-theoretic complexities. Therefore, it cannot be applied to classic models such as inventory management and dynamic pricing models that have continuous state spaces, and students may not be aware of the possible challenges involved in studying dynamic programming models with general state spaces. To address this, we provide conditions and a self-contained simple proof that establish when the principle of optimality for discounted dynamic programming is valid. These conditions shed light on the difficulties that may arise in the general state space case. We provide examples from the literature that include the relatively involved case of universally measurable dynamic programming and the simple case of finite dynamic programming where our main result can be applied to show that the principle of optimality holds.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文