How to align left bracket equation

Question

\begin{equation}
    \left\{
    \begin{matrix}
    Maximiser &L(\alpha)=\sum_{i=1}^{n} \alpha_i -\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_i \alpha_j y_iy_j \textbf{x}^T\textbf{x} &\\ 
    \text{sous contraintes} & & \\ 
    &  \sum_{i=1}^{n}\alpha_iy_i=0 \\
    & \alpha_i>0&
    \end{matrix}
    \right.
\end{equation}

Answer 1

I would like to suggest that (a) the equations be typeset in display style -- done most easily by using a dcases environment -- and (b) the constraint statements be aligned on their respective inequality/equality particles -- using, say, an aligned environment. Oh, and do use \mathbf instead of \textbf while in math mode.

\documentclass{article}
\usepackage{mathtools,mathptmx}
\begin{document}
\begin{equation}
\begin{dcases}
\text{Maximiser} &
L(\alpha)=\sum_{i=1}^{n} \alpha_i 
  -\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_i \alpha_j y_iy_j \mathbf{x}^T\!\mathbf{x} \\ 
\text{sous contraintes} &
\!\begin{aligned}[t]
   \sum_{i=1}^{n}\alpha_iy_i&=0 \\
   \forall\ i\colon\alpha_i&>0
\end{aligned}
\end{dcases}
\end{equation}
\end{document}

Answer 2

I wouldn't go to a new line after “sous contraintes”, but I propose both solutions.

\documentclass{article}
\usepackage{amsmath}
\usepackage{newtxtext,newtxmath}

\DeclareMathOperator{\Maximize}{\mathit{Maximiser}}
\newcommand{\subjto}{\textnormal{sous contraintes}}

\begin{document}

\begin{equation}
\left\{\begin{aligned}
& \Maximize\quad
  L(\alpha)=
    \sum_{i=1}^{n} \alpha_i 
    -\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_i \alpha_j y_iy_j \mathbf{x}^T\mathbf{x} \\
& \begin{aligned}
  \subjto\quad
    & \sum_{i=1}^{n}\alpha_iy_i=0 \\
    & \forall i\colon\alpha_i>0
  \end{aligned}
\end{aligned}\right.
\end{equation}

\begin{equation}
\left\{\begin{aligned}
& \Maximize\quad
  L(\alpha)=
    \sum_{i=1}^{n} \alpha_i 
    -\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_i \alpha_j y_iy_j \mathbf{x}^T\mathbf{x} \\
& \begin{aligned}
  \subjto\quad\\
    & \sum_{i=1}^{n}\alpha_iy_i=0 \\
    & \forall i\colon\alpha_i>0
  \end{aligned}
\end{aligned}\right.
\end{equation}

\end{document}

Better define abstract versions of the operators, so you can easily port documents from one language to the other.

Answer 3

\begin{equation}\label{hypersep}
    \begin{cases}
    Maximiser &L(\alpha)=\sum_{i=1}^{n} \alpha_i -\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_i \alpha_j y_iy_j \textbf{x}^T\textbf{x}\\
    \text{sous contraintes} &\\ 
    &  \sum_{i=1}^{n}\alpha_iy_i=0 \\
    & \alpha_i>0
    \end{cases}
\end{equation}

Answer 4

\left\{
\begin{array}{ll}
    \textrm{Maximiser}
    &
    L(\alpha)=\sum_{i=1}^{n}\alpha_{i}-\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_{i}\alpha_{j}y_{i}y_{j}\textbf{x}^{T}\textbf{x}
    \\
    \textrm{sous contraintes}
    &
    \begin{aligned}[t]
        \sum\nolimits _{i=1}^{n}\alpha_{i}y_{i}
        & 
        =0
        \\
        \alpha_{i}
        &
        >0
    \end{aligned}
\end{array}
\right.

I would add a bit of vertical space between array rows using \\[15pt] instead of simple \\: