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<p>待做。</p>
<p>规划问题包括<strong>线性规划</strong>，<strong>整数规划</strong>和<strong>非线性规划</strong>三大类。</p>
<h2 id="线性规划">线性规划</h2>
<p>线性规划的目标函数可以是求最大值，也可以是求最小值；约束条件的不等号可以是小于号，也可以是大于号。为了避免形式多样性带来的不变，Matlab
中规定线性规划的标准形式为</p>
<p><span class="math display">\[
\mathop{\min}\limits_{\mathbf{x}} \mathbf{c}^\text{T}\mathbf{x} \quad
\text{s.t.}\quad \mathbf{Ax}\le\mathbf{b},
\]</span> 其中，<span class="math inline">\(\mathbf{c}\)</span> 和 <span
class="math inline">\(\mathbf{x}\)</span> 为 <span
class="math inline">\(n\)</span> 维列向量；<span
class="math inline">\(\mathbf{b}\)</span> 为 <span
class="math inline">\(m\)</span> 维列向量；<span
class="math inline">\(\mathbf{A}\)</span> 为 <span
class="math inline">\(m\times n\)</span> 矩阵。</p>
<p>对于线性规划：</p>
<p><span class="math display">\[\mathop{\max}\limits_{\mathbf{x}}
\mathbf{c}^\text{T}\mathbf{x} \quad \text{s.t.}\quad
\mathbf{Ax}\ge\mathbf{b},\]</span></p>
<p>Matlab 标准形式为</p>
<p><span
class="math display">\[\mathop{\min}\limits_{\mathbf{x}}-\mathbf{c}^\text{T}\mathbf{x}\quad\text{s.t.}\quad-\mathbf{Ax}\le-\mathbf{b}.\]</span></p>
<p>Matlab
中线性规划的基本函数形式为<code>linprog(c,A,b)</code>，返回值是向量
<span class="math inline">\(\mathbf{x}\)</span>
的值。对于寻找如下形式的最小值问题：</p>
<p><span
class="math display">\[\mathop{\min}\limits_{\mathbf{x}}f^\text{T}\mathbf{x}\quad\text{s.t.}\begin{cases}\mathbf{A}\cdot\mathbf{x}\le\mathbf{b}\\\mathbf{Aeq}\cdot\mathbf{x}=\mathbf{beq}\\\mathbf{lb}\le\mathbf{x}\le\mathbf{ub}\end{cases},\]</span></p>
<p>可通过</p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">x = linprog(f,A,b,Aeq,beq,lb,ub,options)</span><br></pre></td></tr></table></figure>
<p>求解。</p>
<span id="more"></span>
<h2 id="非线性规划">非线性规划</h2>
<p>非线性规划问题是目标函数或约束函数中包含非线性函数。Matlab
中非线性规划的基本函数形式为<code>fmincon(fun,x0,A,b)</code>，返回值是向量
<span class="math inline">\(\mathbf{x}\)</span>
的值。对于寻找如下形式的最小值问题：</p>
<p><span class="math display">\[
\mathop{\min}\limits_{\mathbf{x}}f(\mathbf{x})\quad\text{s.t.}\begin{cases}c(\mathbf{x})\le\mathbf{0}\\ceq(\mathbf{x})=\mathbf{0}\\\mathbf{A}\cdot\mathbf{x}\le\mathbf{b}\\\mathbf{Aeq}\cdot\mathbf{x}=\mathbf{beq}\\\mathbf{lb}\le\mathbf{x}\le\mathbf{ub}\end{cases},
\]</span> 可通过</p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)</span><br></pre></td></tr></table></figure>
<p>求解。</p>
<h2 id="整数规划">整数规划</h2>
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