[Tex/LaTex] How to put a curly brace in the right of three lines within an align-environment

alignbraces

I have a question concerning how to add a curly brace at the right side of three lines in an align-environment. The following is an example of what I'm speaking about:

\documentclass[a4paper,10pt]{article}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath,amsthm, amsfonts,amssymb, enumerate}

\begin{document}
\begin{align}
&\rlap{\textbf{Optimization model}}
\nonumber \\
&                  && z=\sum_{i\in I} x_i\cdot y_i \cdot z_i \rightarrow \max
\nonumber \\
&\text{s.t.} && x_i \le b_i, 
             &  i\in I, \\
&            && x_i \le y_i,
             &  i \in I,  \\
&            && y_i \le z_i,
             &  i\in I,  \\
&            && z_i \le x_i,
             & i\in I, \\
&            && x_i, y_i, z_i \in \mathbb{R}_{+},
             & i\in I. 
\end{align}
\end{document}

There are five lines of constraints (after the "s.t."). I would like to make a curly brace over the first parts of the second, third and fourth line (i.e. over x_i \le y_i, y_i \le z_i and z_i \le x_i) and write a single "i \in I" in the right of this brace. I hope it was possible to understand it.

Can you please tell my how to achieve this? (The general style of this align-environment should not be changed.)

PS: Here is a picture of how it should look like:
curly brace in the right of the first parts of lines 2 up to 4

Best Answer

With help of mathtools package you can use rcases environment.:

\documentclass[a4paper,10pt]{article}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath,amsthm, amsfonts,amssymb}
\usepackage{mathtools}% <-- new package for rcases

% for show just the equation ...
\usepackage[active,displaymath,tightpage]{preview}
    \setlength\PreviewBorder{1em}


\begin{document}
\begin{align}
&\rlap{\textbf{Optimization model}}
\nonumber \\
&                  && z=\sum_{i\in I} x_i\cdot y_i \cdot z_i \rightarrow \max
\nonumber \\
&\text{s.t.} && x_i \le b_i,                        &  i\in I, \\
&            && %\!\!% first approximation of  misalignment compensation ...
                \negthickspace% for compensation of rcases misalignment
                \begin{rcases}  
                x_i    \le y_i \\
                y_i    \le z_i \\
                z_i    \le x_i
                    \end{rcases}                    & i\in I, \\
&            && x_i, y_i, z_i \in \mathbb{R}_{+},   & i\in I.
\end{align}
\end{document}

Hope that this is what you like to achieve. enter image description here

Edit: Better alignment is achieved if you instead of \!\! you use \negthickspace. It is approximately \mkern-5mu as suggest Bernard in his answer. The picture is replaced with new, where this \negthickspace is used. Use of alignat instead of align is probably more correct selection, but final result is the same.

Usage:

Mark the horizontal position with: \tikzmark{right}. Here I used the end of the header procedure Euclid(a,b) as the right` node. Hence the horizontal position of the brace.
Mark the top of the brace with: \tikzmark{top}.
Mark the bottom of the brace with \tikzmark{bottom}.
Call \AddNote{<top node>}{<bottom node>}{<right node>}{<text>}.

These node names top, bottom, and right are arbitrary so if you have multiple places in the same algorithm where you want to place such notes, you can use different node names. Just pass them to the \AddNote macro.

Notes:

This does require two runs. First one to determine the locations, and the second to do the drawing.

Further Enhancements:

The macro \AddNote could accept additional formatting parameters for the text of the node, and line color selection. Currently this is hard coded to used red and text width=2.5cm.

References:

\tikzmark is from Adding a large brace next to a body of text.

Here are a few similar applications using \tikzmark:

Code:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing,calc}
\newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};}

\newcommand*{\AddNote}[4]{%
    \begin{tikzpicture}[overlay, remember picture]
        \draw [decoration={brace,amplitude=0.5em},decorate,ultra thick,red]
            ($(#3)!(#1.north)!($(#3)-(0,1)$)$) --  
            ($(#3)!(#2.south)!($(#3)-(0,1)$)$)
                node [align=center, text width=2.5cm, pos=0.5, anchor=west] {#4};
    \end{tikzpicture}
}%

\usepackage{algpseudocode}% http://ctan.org/pkg/algorithmicx
\usepackage{algorithm}% http://ctan.org/pkg/algorithm
\begin{document}
\algnewcommand{\algorithmicgoto}{\textbf{go to}}%
\algnewcommand{\Goto}[1]{\algorithmicgoto~\ref{#1}}%
\begin{algorithm}
  \caption{Euclid’s algorithm}\label{euclid}
  \begin{algorithmic}[1]
    \Procedure{Euclid}{$a,b$}\tikzmark{right}\Comment{The g.c.d. of a and b}
    \State $r\gets a\bmod b$
    \While{$r\not=0$} \tikzmark{top}\Comment{We have the answer if r is 0} \label{marker}
      \State $a\gets b$
      \State $b\gets r$
      \State $r\gets a\bmod b$
    \EndWhile \tikzmark{bottom}
    \State \textbf{return} $b$\Comment{The gcd is b}
    \State \Goto{marker}
  \EndProcedure
\end{algorithmic}
\AddNote{top}{bottom}{right}{We loop here until $r=0$.}
\end{algorithm}
\end{document}

[Tex/LaTex] How to put a side brace around several lines in the align* environment

You don't need any additional packages to do this. Here's a way using a regular array:

enter image description here

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\begin{document}
\noindent
\begin{minipage}{.5\textwidth}
\begin{align*}
    f(x) & = a \\
    g(x) & = ax + b \\
    h(x) & = ax^2 + bx + c \\
    i(x) & = ax^3 + bx^2 + cx + d \\
    j(x) & = ax^4 + bx^3 + cx^2 + dx + e
\end{align*}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\[
  \setlength{\jot}{4.5pt}
  \begin{array}{r@{\;}l}
    f(x) & = a \\[\jot]
    g(x) & = ax + b \\[\jot]
    h(x) & = ax^2 + bx + c \hspace*{3em}
      \smash{\left.\begin{array}{@{}c@{}}\\[\jot]\\[\jot]\\[\jot]\end{array}\right\}} \\[\jot]
    i(x) & = ax^3 + bx^2 + cx + d \\[\jot]
    j(x) & = ax^4 + bx^3 + cx^2 + dx + e
  \end{array}
\]
\end{minipage}%
\end{document}

On the left is the regular align* environment (from amsmath), while the right should an array. A zero-height 3-row array with only a right \} delimiter is added to the middle equation to group the output.

Another option, this time using blkarray:

enter image description here

\documentclass{article}
\usepackage{blkarray}% http://ctan.org/pkg/blkarray
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\begin{document}
\noindent
\begin{minipage}{.5\textwidth}
\begin{align*}
    f(x) & = a \\
    g(x) & = ax + b \\
    h(x) & = ax^2 + bx + c \\
    i(x) & = ax^3 + bx^2 + cx + d \\
    j(x) & = ax^4 + bx^3 + cx^2 + dx + e
\end{align*}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\[
  \def\arraystretch{1.1}
  \begin{blockarray}{r@{\;}l}
    f(x) & = a \\[\jot]
    \begin{block}{r@{\;}l\}}
      g(x) & = ax + b \\[\jot]
      h(x) & = ax^2 + bx + c \\[\jot]
      i(x) & = ax^3 + bx^2 + cx + d \\[\jot]
    \end{block}
    j(x) & = ax^4 + bx^3 + cx^2 + dx + e
  \end{blockarray}
\]
\end{minipage}%
\end{document}

You may have to play around with the vertical displacement, compared to the align* environment.