“Minimize” vs. “Min” in Optimization Problems

When writing an optimization problem, write “minimize” and “subject to” instead of “min” and “such that” or “s.t.”.

Correct	Incorrect
\[\begin{aligned} \operatorname{minimize} \quad & f(x) \\ \operatorname{subject to} \quad & Ax = b, \end{aligned} \]	\[ \begin{aligned} \operatorname{min} \quad & f(x) \\ \operatorname{s.t.} \quad& Ax = b. \end{aligned} \]

The problem with writing “$\operatorname{min} f(x)$” instead of “$\operatorname{minimize} f(x)$” is that the expression “$\operatorname{min} f(x)$” represents the minimal value of $f$ whereas “$\operatorname{minimize} f(x)$” is a statement of the goal of the optimization problem.

We use the following code when writing optimization problems. In the preamble, we define \minimize, \maximize, and \subjecto macros, as follows:

% Usage: \minimize{\x \in \reals}
\newcommand*{\minimize}[1]{\operatorname*{minimize}_{#1}\quad} 
% Usage: \maximize{\x \in \reals}
\newcommand*{\maximize}[1]{\operatorname*{maximize}_{#1}\quad} 
\newcommand{\subjectto}{\textup{subject to}\quad}

The starred version of the \operatorname macro causes subscripts to be placed below the text in display equations.

In the document body, we write an optimization problem as

\begin{align*}
    \minimize{x \in \reals^n} & f(x) \\
    \subjectto & Ax \leq 0 \\
               & Bx = 0.
\end{align*}

If space is a concern, you may abbreviate “subject to” as “subj. to” (\textup{subj.\ to}).