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算法的乐趣之动态规划法.html
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算法的乐趣之动态规划法.html
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<span>算法的乐趣之动态规划法</span>
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发布 :
2020-01-22
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算法
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<h1 id="动态规划（Dynamic-Programming）"><a href="#动态规划（Dynamic-Programming）" class="headerlink" title="动态规划（Dynamic Programming）"></a>动态规划（Dynamic Programming）</h1><p>动态规划（Dynamic Programming）是解决多阶段决策问题常用的最优化理论，动态规划和分治法一样，也是通过定义子问题，先求解子问题，然后在由子问题的解组合出原问题的解。它与分治法的区别是动态规划的子问题之间存在堆叠关系（递推关系式确定的递推关系）。</p>
<p>动态规划适合求解多阶段（状态转换）决策问题的最优解，也可用于含有线性或非线性递推关系的最优解问题，但是这些问题都必须满足最优化原理和子问题的“无后向性”。</p>
<ul>
<li><strong>最优化原理</strong>：最优化原理其实就是问题的最优子结构的性质，如果一个问题的最优子结构是不论过去状态和决策如何，对前面的决策所形成的状态而言，其后的决策必须构成最优策略。也就是说，不管之前的决策是否是最优决策，都必须保证从现在开始的决策是在之前决策基础上的最优决策，则这样的最优子结构就符合最优化原理。</li>
<li><strong>无后向性（无后效性）</strong>：所谓“无后向性”，就是当各个阶段的子问题确定以后，对于某个特定阶段的子问题来说，它之前各个阶段的子问题的决策只影响该阶段的决策，对该阶段之后的决策不产生影响。</li>
</ul>
<p>从算法设计的角度分析，使用动态规划法一般需要四个步骤，分别是：</p>
<ul>
<li>1.定义最优子问题（最优解的子结构）</li>
<li>2.定义状态（最优解的值）</li>
<li>3.定义决策和状态转换方程（定义计算最优解的值的方法）</li>
<li>3.确定边界条件</li>
</ul>
<h2 id="1-定义最优子问题"><a href="#1-定义最优子问题" class="headerlink" title="1.定义最优子问题"></a>1.定义最优子问题</h2><p>定义最优子问题，也就是最优解的子结构，它确定问题的优化目标以及如何决策最优解，并对决策过程划分阶段。所谓阶段，可以理解为一个问题从开始到解决需要经过的环节，这些环节前后关联。阶段划分以后，对问题的求解就变成了各个阶段分别进行最优化决策，问题的解就变成按照阶段顺序依次选择的一个决策序列。这就需要先定义状态，有了状态的定义，只要状态发生了变化，就可以认为是一个阶段。</p>
<h2 id="2-定义状态"><a href="#2-定义状态" class="headerlink" title="2.定义状态"></a>2.定义状态</h2><p>状态既是决策的对象，也是决策的结果，对于每个阶段来说，对起始状态施加决策，使得状态发生改变，得到决策的结果状态。初始状态经过每一个阶段的决策（状态改变）之后，最终得到的状态就是问题的解。当然，不是所有的决策序列施加于初始状态后都可以得到最优解，只有一个决策序列能得到最优解。</p>
<h3 id="3-定义决策和状态转换方程"><a href="#3-定义决策和状态转换方程" class="headerlink" title="3.定义决策和状态转换方程"></a>3.定义决策和状态转换方程</h3><p>决策就是能使状态发生转变的选择动作，如果选择动作有多个，则决策就是取其中能使得阶段结果最优的那一个。状态转换方程是描述状态转换关系的一系列等式，也就是从 n-1 阶段到 n 阶段演化的规律。状态转换取决于子问题的堆叠方式，如果状态定义得不合适，会导致子问题之间没有重叠，也就不存在状态转换关系了。没有状态转换关系，动态规划也就没有意义了，实际算法就退化为像分治法那样的朴素递归搜索算法了。</p>
<h3 id="4-确定边界条件"><a href="#4-确定边界条件" class="headerlink" title="4.确定边界条件"></a>4.确定边界条件</h3><p>对于递归加备忘录方式（记忆搜索）实现的动态规划方法，边界条件实际上就是递归终结条件，无需额外的计算。对于使用递推关系直接实现的动态规划方法，需要确定状态转换方程的递推式的初始条件或边界条件，否则无法开始递推计算。</p>
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