小高分享（90）运筹学中的动态规划简介

摘要：动态规划是运筹学中一种重要的数学方法，主要用于解决具有重叠子问题和最优子结构性质的多阶段决策问题。它将复杂问题分解为一系列相互关联的子问题，通过求解这些子问题来逐步得到原问题的最优解。例如，在资源分配、生产调度等问题中，动态规划能够帮助我们找到在有限资源下实现

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小高分享（90）运筹学中的动态规划简介

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A Brief Introduction to dynamic programming in Operations Research by Xiao Gao (90)

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一、基本概念

First, the basic concept

动态规划是运筹学中一种重要的数学方法，主要用于解决具有重叠子问题和最优子结构性质的多阶段决策问题。它将复杂问题分解为一系列相互关联的子问题，通过求解这些子问题来逐步得到原问题的最优解。例如，在资源分配、生产调度等问题中，动态规划能够帮助我们找到在有限资源下实现目标最优的方案。它不是一种具体的算法，而是一种思想，通过记录子问题的解来避免重复计算，从而提高求解效率。

Dynamic programming is an important mathematical method in operations research, which is mainly used to solve multi-stage decision-making problems with overlapping subproblems and optimal substructures. It decomposes a complex problem into a series of interrelated sub-problems, and solves these sub-problems to gradually obtain the optimal solution of the original problem. For example, in resource allocation, production scheduling, and other problems, dynamic programming can help us find the optimal solution to achieve the goal with limited resources. It is not a specific algorithm, but an idea that improves the efficiency of solving by recording the solution of sub-problems to avoid double counting.

二、基本方程

Second, the basic equation

动态规划的基本方程是解决问题的关键。它通常表示为 ( f_k(x_k) = \min_{u_k} { c_k(x_k, u_k) + f_{k+1}(x_{k+1}) } )，其中 ( f_k(x_k) ) 表示在第 ( k ) 阶段状态为 ( x_k ) 时的最优指标函数值，( u_k ) 是决策变量，( c_k(x_k, u_k) ) 是当前阶段的成本或收益，( x_{k+1} ) 是下一阶段的状态。这个方程体现了动态规划的自底向上求解思路，从最后一个阶段开始，逐步向前求解，直到得到初始阶段的最优解。The basic equations of dynamic programming are the key to solving the problem. It is usually expressed as ( f_k(x_k) = min_{u_k} { c_k(x_k, u_k) f_{k 1}(x_{k 1}) } where ( f_k( x_k) ) represents the optimal metric function value when the state is ( x_k ) in the ( k ) stage, ( u_k ) is the decision variable, ( c_k( x_k, u_k) ) ) is the cost or benefit of the current stage, and ( x_{k 1} ) is the state of the next stage. This equation embodies the bottom-up solution idea of dynamic programming, starting from the last stage and gradually moving forward until the optimal solution of the initial stage is obtained.

三、最优化原理

Third,optimization principle

最优化原理是动态规划的核心。它指出，一个过程的最优策略具有这样的性质，即无论过去的状态和决策如何，相对于前面的决策所形成的状态而言，余下的决策序列必须构成一个最优策略。这意味着在求解过程中，我们只需要关注当前阶段和下一阶段的状态与决策，而不必考虑之前的过程，大大简化了问题的复杂性。

The principle of optimization is at the heart of dynamic programming. It states that the optimal strategy of a process has the property that the remaining sequence of decisions must constitute an optimal strategy relative to the state formed by the previous decision, regardless of the past state and decision. This means that we only need to focus on the state and decisions of the current and next stages during the solution process, rather than the previous process, greatly simplifying the complexity of the problem.

四、动态规划应用

Fourth, dynamic planning applications

动态规划在各个领域都有广泛的应用。在金融领域，可用于投资组合优化，帮助投资者在风险和收益之间找到平衡；在物流领域，可用于路径规划，减少运输成本和时间；在计算机科学中，可用于算法设计，如字符串匹配、图像处理等。通过动态规划，我们能够解决许多传统方法难以处理的复杂问题，提高决策的科学性和效率。

Dynamic programming has a wide range of applications in various fields. In the financial sector, it can be used for portfolio optimization to help investors find a balance between risk and return; In the field of logistics, it can be used for path planning to reduce transportation costs and time; In computer science, it can be used for algorithm design, such as string matching, image processing, etc. Through dynamic programming, we can solve many complex problems that are difficult to deal with with traditional methods, and improve the scientific and efficient decision-making.

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