Wrong font family in algorithm2e

The correct command is \mathrm in every case.

\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage[ruled]{algorithm2e}

\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}

\begin{document}

\begin{algorithm}
\DontPrintSemicolon 

\KwIn{Blalalala.}
\KwOut{Blalalala}

\For{$t = 1$ to $T$}{
  Fit $\Psi_{t+1}$ to $\mathcal{H}_{t}$\\ 
  $\mathbf{x} \leftarrow \argmax_{\mathbf{x} \in \mathcal{X}} 
    a(\mu (\Psi_{t+1} (\mathbf{x})) , \sigma(\Psi_{t+1} (\mathbf{x})) , J^{\textup{min}})$\\
  Check $c=J(\mathbf{x})$\\
  $\mathcal{H}_{t+1} \leftarrow \mathcal{H}_{t} \cup \lbrace (\mathbf{x},c) \rbrace $\\
  \If{ $c (\mathbf{x}) < c^{\textup{min}} $}{
    $\mathbf{x}^{\textup{min}},c^{\textup{min}} \leftarrow \mathbf{x},c$
  }
}
\Return $\mathbf{x}^{\textup{min}}$

\caption{Testing}
\end{algorithm}

\end{document}

You'd get the same result with \textup. The problem is that \text picks up the font that's current when the formula starts, which is italic in the argument to \If.

Side note. Your MWE cannot produce “mín” from \min. That happens if you load babel-spanish, for instance.

At least, I got a workaround.

\documentclass{article}
\usepackage{amsmath,amssymb}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
\usepackage[ruled]{algorithm2e}
\begin{document}
    \begin{algorithm}
        \DontPrintSemicolon
        \KwIn{Blalalala.}
        \KwOut{Blalalala}
        \For{$t = 1$ to $T$}{
            Fit $\Psi_{t+1}$ to $\mathcal{H}_{t}$\\
            $\mathbf{x}\leftarrow \argmax_{\mathbf{x}\in\mathcal{X}} a(\mu (\Psi_{t+1} (\mathbf{x})),\sigma(\Psi_{t+1} (\mathbf{x})) , J^\text{min})$\\
            Check $c=J(\mathbf{x})$\\
            $\mathcal{H}_{t+1} \leftarrow \mathcal{H}_{t} \cup \left\lbrace(\mathbf{x},c)\right\rbrace$\\
            \If{ $c (\mathbf{x}) < c^\mathrm{min} $}{
                $\mathbf{x}^\text{min},c^\text{min} \leftarrow \mathbf{x},c $
            }
        }
        \Return $\mathbf{x}^\text{min}$
        \caption{Testing}
    \end{algorithm}
\end{document}

Surely, there are fonts which make a difference between \text{} and \mathrm{} visible to the naked eye, but for this example, I cannot distinguish my output "min"s whether it was created with the further or the latter.