Totui

如果用LaTeX去寫最佳化問題(Optimization Problem)？

因為之前學長都是把最佳化問題做成一張圖檔，

然後直接貼在paper上，

並且有些圖檔在轉化的時候品質變得很差。

所以整理了一下別人的內容如下：

Constrained optimization problems are almost everywhere in engineering research.

A mathematical description of those problems with a single objective is

to minimize or maximize an objective function

over a set of decision variables under a set of constraints.

There are different ways to format optimization problems;

personally, I follow the format used in the book “Convex Optimization”

by Stephen Boyd and Lieven Vandenberghe.

For example, a general optimization problem has the form
   

This is generated by the following  code:

\begin{equation*}
  \begin{aligned}
    & \underset{x}{\text{minimize}}
    & & f_0(x) \\
    & \text{subject to}
    & & f_i(x) \leq b_i, \; i = 1, \ldots, m.
  \end{aligned}
\end{equation*}

As seen in the code, the formatting is done by the aligned environment,

which is defined in the amsmath package, so you need to include the following line in the preamble:

Unlike the tabular environment, in which you can specify the alignment of each column,

in the aligned environment, each column (separated by &) has a default alignment,

which alternates between right and left-aligned.

Therefore, all the odd columns are right-aligned and all the even columns are left-aligned.

We conclude with a real example:

The above problem is formulated for completing low-rank positive semidefinite matrices.

It is convex, or more precisely, it is a semidefinite program.  The corresponding  code is

\begin{equation*}
  \begin{aligned}
    & \underset{X}{\text{minimize}}
    & & \mathrm{trace}(X) \\
    & \text{subject to}
    & & X_{ij} = M_{ij}, \; (i,j) \in \Omega, \\
    &&& X \succeq 0.
  \end{aligned}
\end{equation*}