How to call command with optional arg using {} instead of []

Question

This is a question re: Lost global definition after \ifthenelse using package xifthen. I'll reference the Answer given in this question here since it's clear and clean.

MWE:

\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{xifthen}

\newcommand{\test}[1][]{%
\ifthenelse{\equal{#1}{}}{omitted}{given}%
}

\begin{document}
\par The optional argument was \test[].
\par The optional argument was \test[shubidu].
\end{document}

Problem:

How to rewrite the command definition of \test so that:

\test{}

Output: omitted

\test{shubidu}

Output: given

EDIT:

Thank you everyone for your tips and tricks. For some more information that may help clarify the purpose of this question: I'm trying to define optional args using {} instead of [] because I'm in the process of making a TeX -> LyX transfer. LyX inserts an optional arg with [] if you click "Insert" on a drop-down menu and type the arg. Otherwise, LyX will insert your arg with {} if you do what I've done below. I would like to keep the formatting of what I've done below which is written with this: \newcommand{\BriefName}[1]{\xdef\BriefName{#1}} but make the argument inserted for \BriefName optional. Thus, if the LyX user enters nothing in the \BriefName line, the program still compiles, with an "empty" argument for the \BriefName command.

Updated MWE:

This is more what I'm looking for. I created this thanks to this Q&A.

\documentclass{article}
\usepackage{xifthen}

\newcommand{\BriefName}[1]{ %
        \xdef\BriefName{#1}\ifx\BriefName\empty
          {}%
        \else
          {#1}%
        \fi
}%

\begin{document}
\par This should be empty:\BriefName{}
\par This should also be empty: \BriefName
\par This should say Argument: \BriefName{Argument}
\par This should also say Argument: \BriefName
\end{document}

Answer 1

You can do it with xparse; the trick is not redefining \BriefName as you're trying to do, but another control sequence holding the current value (empty at the beginning).

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\BriefName}{g}
 {
  \IfNoValueF{#1}
   {
    \tl_gset:Nn \g_jalep_briefname_tl { #1 }
   }
  \tl_use:N \g_jalep_briefname_tl
 }
\tl_new:N \g_jalep_briefname_tl
\ExplSyntaxOff

\begin{document}

This should be empty:\BriefName

This should also be empty: \BriefName[]

This should say Argument: \BriefName{Argument}

This should also say Argument: \BriefName

\end{document}

However, the standard LaTeX syntax uses [] for optional arguments, so it would be much better to stick with it.

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\BriefName}{o}
 {
  \IfNoValueF{#1}
   {
    \tl_gset:Nn \g_jalep_briefname_tl { #1 }
   }
  \tl_use:N \g_jalep_briefname_tl
 }
\tl_new:N \g_jalep_briefname_tl
\ExplSyntaxOff

\begin{document}

This should be empty:\BriefName

This should also be empty: \BriefName[]

This should say Argument: \BriefName[Argument]

This should also say Argument: \BriefName

\end{document}

The version with [] also admits a solution with classical tools.

\documentclass{article}

\newcommand{\BriefName}[1][]{%
  \if\relax\detokenize{#1}\relax
  \else
    \gdef\jalepbriefname{#1}%
  \fi
  \jalepbriefname
}
\def\jalepbriefname{}

\begin{document}

This should be empty:\BriefName

This should also be empty: \BriefName[]

This should say Argument: \BriefName[Argument]

This should also say Argument: \BriefName

\end{document}

Answer 2

To get what you ask you just need to delete [] in the declaration

\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{xifthen}

\newcommand{\test}[1]{%
\ifthenelse{\equal{#1}{}}{omitted}{given}%
}

\begin{document}

The optional argument was \test{}.

The optional argument was \test{shubidu}.

\end{document}

However the original example appears to have been mis-using the optional argument syntax.

The intended syntax for the "not given" case is \test not \test[]

So you do not need to test for an empty argument, latex has already tested for [ being present:

\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{xifthen}

\newcommand{\test}[1][default]{#1\ldots}

\begin{document}

The optional argument was \test.

The optional argument was \test[shubidu].

\end{document}