\documentclass{article}

\usepackage{algorithm2e,enumerate,mathtools}
\SetKwFor{KWHILE}{for}{do}{endfor}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}

\begin{document}

\begin{algorithm}[H]
  \caption{Algorithm caption}
  \KWHILE{stop condition}{
    \begin{enumerate}[(a)]
      \item sparse coding: problem
        \[
          \min_{\mathbf{x}_i} \norm[\Big]{ \mathbf{y}_i - \mathbf{D}^{(J-1)} \mathbf{x}_i}_2^2
          \quad \text{s.t.} \quad
          \norm{ \mathbf{x}_i }_0 \leq s
        \]
      \item dictionary update: \ldots

        \makebox[\linewidth]{$\displaystyle
          \min_{\mathbf{x}_i} \norm[\Big]{ \mathbf{y}_i - \mathbf{D}^{(J-1)} \mathbf{x}_i}_2^2
          \quad \text{s.t.} \quad
          \norm{ \mathbf{x}_i }_0 \leq s
        $}

      \item dictionary update: \ldots
    \end{enumerate}}
\end{algorithm}

\end{document}