# “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}`

).