\operatorname
is not the right tool for this job. You might do
\newcommand{\smoothmin}[1]{%
\operatorname*{min_{\mathgroup=-1 #1}}%
}
but an input such as \smoothmin{P,-1}_{i=0,1,\dots,n}
would produce

because minus signs become hyphens in the argument to \operatorname*
.
Fix:
\documentclass{article}
\usepackage{amsmath}
\newcommand{\smoothmin}[1]{\mathop{\min\nolimits_{#1}}}
\begin{document}
\[
\smoothmin{P,0}_{i=0,\dots,n}
\]
\end{document}

A suggestion for being able to use this also in inline formulas:
\documentclass{article}
\usepackage{amsmath}
\makeatletter
\NewDocumentCommand{\smoothmin}{me{_}}{%
\mathop{\mathpalette\smoothmin@{{#1}{#2}}}%
}
\NewDocumentCommand{\smoothmin@}{mm}{\smoothmin@@#1#2}
\NewDocumentCommand{\smoothmin@@}{mmm}{%
% #1 = math style, #2 = subscript, #3 = optional condition
\ifx#1\displaystyle
\mathop{\min\nolimits_{#2}}\IfValueT{#3}{_{#3}}%
\else
\min_{#2\IfValueT{#3}{;\,#3}}%
\fi
}
\makeatother
\begin{document}
\[
\smoothmin{P,0}_{i=0,\dots,n}a_i
\]
\begin{center}
$\smoothmin{P,0}_{i=0,\dots,n}a_i$
\end{center}
\end{document}

\newcommand{\smoothmin}[1]{\min_{#1}}
along with${\smoothmin{P,0}}_{i=0,\dots,n}$
? Or, if you prefer,\newcommand{\smoothmin}[1]{\operatorname*{min}_{#1}}
along with${\smoothmin{P,0}}_{i=0,\dots,n}$
?\operatorname
instead of\operatorname*
.