☰

优化方程的格式

tag-icon • tag-icon equations align subequations
优化方程的格式

我正在为两栏科学论文写一个优化问题。问题是我使用以下代码来生成方程，但输出的美观度不佳，因为它在列的左侧留下了很多空间。有没有比使用下面提到的代码更好的方法（除了子方程）来编写这些方程。此外，我想在一行中编写一个方程（约束），其中包含所有变量（如果可能）。有什么方法可以实现这两个目标？以下是我工作的 MWE。

\documentclass[conference]{IEEEtran}
\usepackage{amsmath}
\begin{document}
\begin{subequations}
\begin{alignat}{2}
& \textbf{P2}\ \min_{\textbf{X}}  &\qquad& f(\textbf{X})\label{eq:OF2}\\
&\text{subject to} &      & Tr(\mathbf{\Psi}_{P,k}^\varphi \textbf{X}) + P_{l_k}^\kappa = 0, \nonumber\\
&                  &      & \hspace{5em}\forall \varphi \in \phi_{k}, \kappa \in \psi_{k}, k \in N\backslash G \label{eq:PBL2}\\
&                  &      & Tr(\mathbf{\Psi}_{Q,k}^\varphi \textbf{X}) + Q_{l_k}^\kappa-y_{c_k}^\kappa Tr(\mathbf{\Psi}_{V,k}^\varphi \textbf{X}) = 0,\nonumber\\
&                  &      & \hspace{5em} \forall \varphi \in \phi_{k}, \kappa \in \psi_{k}, k \in N\backslash G \label{eq:QBL2}\\
&                  &      & \underline{P}_{g_i} \leq Tr(\mathbf{\Psi}_{P,i}^\varphi \textbf{X}) + P_{l_i}^\kappa \leq \overline{P}_{g_i}, \nonumber \\
&                  &      & \hspace{5em} \forall \varphi \in \phi_{i}, \kappa \in \psi_{i}, i \in G \label{eq:PBG2}\\
&                  &      & \underline{Q}_{g_i} \leq Tr(\mathbf{\Psi}_{Q,i}^\varphi \textbf{X}) + Q_{l_i}^\kappa \leq \overline{Q}_{g_i},\nonumber\\
&                  &      & \hspace{5em} \forall \varphi \in \phi_{i}, \kappa \in \psi_{i}, i \in G \label{eq:QBG2}\\
&                  &      & (\underline{V}_k)^2 \leq Tr(\mathbf{\Psi}_{V,k}^\varphi \textbf{X}) \leq (\overline{V_k})^2,\nonumber \\
&                  &      &\hspace{5em}\forall \varphi \in \eta_{k}, k \in N^+,\label{eq:Vol2}\\
&                  &      & [\textbf{X}]_{\phi_{0}\times \phi_{0}} = \textbf{v}_0\textbf{v}_0^\textit{H}\\
&                  &      & \textbf{X}\succeq 0,\\
&                  &      & rank(\textbf{X})=1 \label{rank_con}
\end{alignat}
\end{subequations}
\end{document}

答案1

这是一个选项，但是列宽对于您的某个限制来说太窄了：

  • 使用\DeclareMathOperatorforTr和rank;
  • 使用\mathbf代替 来\textbf表示b旧的facemath内容。或者，有\boldsymbol或\bm（来自bm（英文）：
  • 用于{+}消除常规操作符周围的间距...在紧要关头；
  • 写出问题描述以打破对齐链接，从而在空间方面提供更多的灵活性

在此处输入图片描述

\documentclass[conference]{IEEEtran}

\usepackage{amsmath}

\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\rank}{rank}

\newcommand{\tightermath}{%
  \setlength{\thickmuskip}{.3\thickmuskip}
  \setlength{\medmuskip}{.3\medmuskip}
  \setlength{\thinmuskip}{.3\thinmuskip}
}
\begin{document}

\begin{subequations}
  \begin{alignat}{2}
    & \textbf{P2}\ \min_{\textbf{X}}  &\qquad& f(\textbf{X}) \\
    &\text{subject to} &      & \Tr(\mathbf{\Psi}_{P,k}^\varphi \textbf{X}) + P_{l_k}^\kappa = 0, \nonumber\\
    &                  &      & \hspace{5em}\forall \varphi \in \phi_{k}, \kappa \in \psi_{k}, k \in N\backslash G \\
    &                  &      & \Tr(\mathbf{\Psi}_{Q,k}^\varphi \textbf{X}) + Q_{l_k}^\kappa-y_{c_k}^\kappa \Tr(\mathbf{\Psi}_{V,k}^\varphi \textbf{X}) = 0, \nonumber\\
    &                  &      & \hspace{5em} \forall \varphi \in \phi_{k}, \kappa \in \psi_{k}, k \in N\backslash G \\
    &                  &      & \underline{P}_{g_i} \leq \Tr(\mathbf{\Psi}_{P,i}^\varphi \textbf{X}) + P_{l_i}^\kappa \leq \overline{P}_{g_i}, \nonumber \\
    &                  &      & \hspace{5em} \forall \varphi \in \phi_{i}, \kappa \in \psi_{i}, i \in G \\
    &                  &      & \underline{Q}_{g_i} \leq \Tr(\mathbf{\Psi}_{Q,i}^\varphi \textbf{X}) + Q_{l_i}^\kappa \leq \overline{Q}_{g_i}, \nonumber\\
    &                  &      & \hspace{5em} \forall \varphi \in \phi_{i}, \kappa \in \psi_{i}, i \in G \\
    &                  &      & (\underline{V}_k)^2 \leq \Tr(\mathbf{\Psi}_{V,k}^\varphi \textbf{X}) \leq (\overline{V_k})^2, \nonumber \\
    &                  &      &\hspace{5em}\forall \varphi \in \eta_{k}, k \in N^+, \\
    &                  &      & [\textbf{X}]_{\phi_{0}\times \phi_{0}} = \textbf{v}_0\textbf{v}_0^\textit{H}\\
    &                  &      & \textbf{X}\succeq 0,\\
    &                  &      & \rank(\textbf{X})=1 \label{rank_con}
  \end{alignat}
\end{subequations}

\newpage

Consider the optimisation \textbf{P2} with the objective to

\begin{subequations}
  \begin{equation}
    \underset{\mathbf{X}}{\text{minimize}}\ f(\mathbf{X})
  \end{equation}
subject to
  \begin{flalign}
    & \Tr(\mathbf{\Psi}_{P,k}^\varphi \mathbf{X}) + P_{l_k}^\kappa = 0, 
      \forall \varphi \in \phi_k, \kappa \in \psi_k, k \in N \setminus G \\
    & \Tr(\mathbf{\Psi}_{Q,k}^\varphi \mathbf{X}) + Q_{l_k}^\kappa-y_{c_k}^\kappa Tr(\mathbf{\Psi}_{V,k}^\varphi \mathbf{X}) = 0, \nonumber \\
    & \hspace{5em} \forall \varphi \in \phi_k, \kappa \in \psi_k, k \in N \setminus G \\
    & \underline{P}_{g_i} \leq \Tr(\mathbf{\Psi}_{P,i}^\varphi \mathbf{X}) {+} P_{l_i}^\kappa \leq \overline{P}_{g_i}, 
      \forall \varphi \in \phi_i, \kappa \in \psi_i, i \in G \\
    & \underline{Q}_{g_i} \leq \Tr(\mathbf{\Psi}_{Q,i}^\varphi \mathbf{X}) {+} Q_{l_i}^\kappa \leq \overline{Q}_{g_i},
      \forall \varphi \in \phi_i, \kappa \in \psi_i, i \in G \\
    & (\underline{V}_k)^2 \leq \Tr(\mathbf{\Psi}_{V,k}^\varphi \mathbf{X}) \leq (\overline{V_k})^2,
      \forall \varphi \in \eta_k, k \in N^+, \\
    & [\mathbf{X}]_{\phi_0 \times \phi_0} = \mathbf{v}_0 \mathbf{v}_0^H \\
    & \mathbf{X} \succeq 0, \\
    & \rank(\mathbf{X}) = 1
  \end{flalign}
\end{subequations}

\end{document}

答案2

您可以使用该optdef包：

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{geometry}
\usepackage{amsmath}
\DeclareMathOperator\Tr{Tr}
\DeclareMathOperator\rank{rank}
\usepackage{optidef}

\begin{document}

\begin{mini!}|l|[0]
    {\mathbf{X}}{f(\mathbf{X})}
    {}
    {\label{eq:OF2}}{}
    \addConstraint{}{ \Tr(\mathbf{\Psi}_{P,k}^\varphi \textbf{X}) + P_{l_k}^\kappa = 0,}{\quad\forall \varphi \in \phi_{k}, \kappa \in \psi_{k}, k \in N\setminus G \label{eq:PBL2}}
      \addConstraint{}{\Tr(\mathbf{\Psi}_{Q,k}^\varphi \mathbf{X}) + Q_{l_k}^\kappa-y_{c_k}^\kappa Tr(\mathbf{\Psi}_{V,k}^\varphi \mathbf{X}) = 0,}{ \quad\forall \varphi \in \phi_{k}, \kappa \in \psi_{k}, k \in N\setminus G\label{eq:QBL2}}
     \addConstraint{}{\:\underline{P\!}_{\mkern1mu g_i} \leq \Tr(\mathbf{\Psi}_{P,i}^\varphi \mathbf{X}) + P_{l_i}^\kappa \leq \overline{P}_{g_i},} {\quad \forall \varphi \in \phi_{i}, \kappa \in \psi_{i}, i \in G \label{eq:PBG2}}
     \addConstraint{}{\, \underline{Q}_{g_i} \leq Tr(\mathbf{\Psi}_{Q,i}^\varphi \mathbf{X}) + Q_{l_i}^\kappa \leq \overline{Q}_{g_i},} {\quad \forall \varphi \in \phi_{i}, \kappa \in \psi_{i}, i \in G \label{eq:QBG2}}
     \addConstraint{}{\,\underline{V}_k)^2 \leq \Tr(\mathbf{\Psi}_{V,k}^\varphi \mathbf{X}) \leq (\overline{V_k})^2,} {\quad\forall \varphi \in \eta_{k}, k \in N^+\label{eq:Vol2}}
     \addConstraint{}{\mkern1.5mu[\mathbf{X}]_{\phi_{0}\times \phi_{0}} = \mathbf{v}_0\mathbf{v}_0^\mathit{H}}
     \addConstraint{}{\mkern1.5mu[\mathbf{X}]\succeq 0}
     \addConstraint{}{\rank(\mathbf{X})=1 \label{rank_con}}
\end{mini!}

\end{document}

在此处输入图片描述

答案3

找到合适的东西非常困难，但也许

在此处输入图片描述

\documentclass[conference]{IEEEtran}
\usepackage{amsmath}
\DeclareMathOperator\Tr{Tr}
\DeclareMathOperator\rank{rank}
\begin{document}
\begin{subequations}
\begin{equation}
\mathbf{P2}\; \min_{\mathbf{X}}  \qquad f(\mathbf{X})\label{eq:OF2}\\
\end{equation}
subject to
\begin{flalign}
&\scriptstyle\Tr(\mathbf{\Psi}_{P,k}^\varphi \mathbf{X}) + P_{l_k}^\kappa = 0,
\quad \forall \varphi \in \phi_{k}, \kappa \in \psi_{k}, k \in N\backslash G \hfill X\label{eq:PBL2}\\
&\scriptstyle\Tr(\mathbf{\Psi}_{Q,k}^\varphi \mathbf{X}) + Q_{l_k}^\kappa-y_{c_k}^\kappa \Tr(\mathbf{\Psi}_{V,k}^\varphi \mathbf{X}) = 0,
\quad \forall \varphi \in \phi_{k}, \kappa \in \psi_{k}, k \in N\backslash G \label{eq:QBL2}\\
&\scriptstyle\underline{P}_{g_i} \leq \Tr(\mathbf{\Psi}_{P,i}^\varphi \mathbf{X}) + P_{l_i}^\kappa \leq \overline{P}_{g_i}, 
\quad \forall \varphi \in \phi_{i}, \kappa \in \psi_{i}, i \in G \label{eq:PBG2}\\
&\scriptstyle\underline{Q}_{g_i} \leq \Tr(\mathbf{\Psi}_{Q,i}^\varphi \mathbf{X}) + Q_{l_i}^\kappa \leq \overline{Q}_{g_i},
\quad \forall \varphi \in \phi_{i}, \kappa \in \psi_{i}, i \in G \label{eq:QBG2}\\
 &\scriptstyle(\underline{V}_k)^2 \leq \Tr(\mathbf{\Psi}_{V,k}^\varphi \mathbf{X}) \leq (\overline{V_k})^2,
\quad\forall \varphi \in \eta_{k}, k \in N^+,\label{eq:Vol2}\\
&\scriptstyle [\mathbf{X}]_{\phi_{0}\times \phi_{0}} = \mathbf{v}_0\mathbf{v}_0^\mathit{H}\\
&\scriptstyle\mathbf{X}\succeq 0,\\
&\scriptstyle\rank(\mathbf{X})=1 \label{rank_con}
\end{flalign}
\end{subequations}
\end{document}

请注意，除了对齐之外， \mathbf不要\textbf使用数学粗体字母，也不要对 rank 或 Tr 等运算符使用数学斜体，字体的设计目的是使相邻的字母看起来不像一个单词，而像单字母变量的乘积。

相关内容