Aligning large Equation with &

Question

I have to use a line break in my equation. But the content in the extra line is squashed to the right. But If I try aligning it with "&" all the content gets massively pushed to the right. Here is the code without alignment:

\begin{align*}
  \phi_{ij}^{(k+1)} =\underset{\phi_{ij} \in \mathbb{R}^k}{\argmax} \langle \phi_{ij}, (\nabla\cdot(2u^{(k+1)}-u^{(k)}))_{ij} - (\nabla_G\cdot(2w^{(k+1)}-w^{(k)}))_{ij} - \lambda^0_{ij}+\lambda^1_{ij}\rangle \\
   - \frac{1}{2\tau}(\phi_{ij} -\phi_{ij}^{(k)})^2  \\
   \Rightarrow 0 \overset{!}{=} (\nabla\cdot(2u^{(k+1)}-u^{(k)}))_{ij} - (\nabla_G\cdot(2w^{(k+1)}-w^{(k)}))_{ij} - \lambda^0_{ij}+\lambda^1_{ij} \\
   + \frac{1}{\tau}(\phi_{ij} - \phi^{(k)}_{ij}) \\
   \phi^{(k+1)}_{ij}:=\phi_{ij} = \phi_{ij}^{(k)} + \tau ((\nabla\cdot(2u^{(k+1)}-u^{(k)}))_{ij} \\
   - (\nabla_G\cdot(2w^{(k+1)}-w^{(k)}))_{ij} - \lambda^0_{ij}+\lambda^1_{ij})
\end{align*}

This is the best I can do. But in red I marked, what I actually want to do.

Answer 1

You can use the split environment within the align* environment along with ampersand(&).

\documentclass{article}

\usepackage{amsmath}
\usepackage{amsfonts}
\DeclareMathOperator*{\argmax}{arg\,max} %Assuming, as not defined in question 


\begin{document}
    \begin{align*}
        \begin{split}
        \phi_{ij}^{(k+1)} ={}& \underset{\phi_{ij} \in \mathbb{R}^k}{\argmax} \langle \phi_{ij}, (\nabla\cdot(2u^{(k+1)}-u^{(k)}))_{ij} - (\nabla_G\cdot(2w^{(k+1)}-w^{(k)}))_{ij} - \lambda^0_{ij}+\lambda^1_{ij}\rangle \\
        & - \frac{1}{2\tau}(\phi_{ij} -\phi_{ij}^{(k)})^2  \\
        \end{split}\\
        \begin{split}
         \Rightarrow 0 \overset{!}{=}{}& (\nabla\cdot(2u^{(k+1)}-u^{(k)}))_{ij} - (\nabla_G\cdot(2w^{(k+1)}-w^{(k)}))_{ij} - \lambda^0_{ij}+\lambda^1_{ij} \\
        &+ \frac{1}{\tau}(\phi_{ij} - \phi^{(k)}_{ij}) \\
        \end{split}\\
        \begin{split}
        {}&\phi^{(k+1)}_{ij}:=\phi_{ij} = \phi_{ij}^{(k)} + \tau ((\nabla\cdot(2u^{(k+1)}-u^{(k)}))_{ij} \\
        &- (\nabla_G\cdot(2w^{(k+1)}-w^{(k)}))_{ij} - \lambda^0_{ij}+\lambda^1_{ij})
        \end{split}
    \end{align*}
    
\end{document}

Answer 2

Edit:

Since your page layout is unknown, it is difficult to suggest a solution for your problem. For simple move second lines to the left at use of article document class with default page layout your equation protrude oft of text area:

By using article document class, mathtools package, align math environment and standard syntax for adding arguments to math functions:

\documentclass{article}
\usepackage{mathtools, amssymb}
\DeclareMathOperator*{\argmax}{arg\,max}

%---------------- Show page layout. Don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%

\begin{document}
    \begin{align*}
\phi_{ij}^{(k+1)} 
     = {}   &  \argmax_{\phi_{ij} \in \mathbb{R}^k} 
                \Bigl\langle \phi_{ij},
                \Bigl(\nabla\cdot\bigl(2u^{(k+1)}-u^{(k)}\bigr)\Bigr)_{ij}
                - \Bigl(\nabla_G\cdot\bigl(2w^{(k+1)}-w^{(k)}\bigr)\Bigr)_{ij}
                - \lambda^0_{ij}+\lambda^1_{ij}
                \Bigr\rangle \\
            & - \frac{1}{2\tau}\bigl(\phi_{ij} - \phi_{ij}^{(k)}\bigr)^2    \\[2ex]
%
\Rightarrow 0 \overset{!}{=}{}
            & \Bigl(\nabla\cdot\bigl(2u^{(k+1)}-u^{(k)}\bigr)\Bigr)_{ij} 
                - \Bigl(\nabla_G\cdot\bigl(2w^{(k+1)}-w^{(k)}\bigr)\Bigr)_{ij} - \lambda^0_{ij}+\lambda^1_{ij} \\
            & + \frac{1}{\tau}\bigl(\phi_{ij} - \phi^{(k)}_{ij}\bigr)       \\[2ex]
%
\phi^{(k+1)}_{ij}
\coloneqq {}&   \phi_{ij} = \phi_{ij}^{(k)} 
                + \tau \Bigl(\bigl(\nabla\cdot(2u^{(k+1)}-u^{(k)}\bigr)\Bigr)_{ij} \\
            & - \Bigl(\nabla_G\cdot\bigl(2w^{(k+1)}-w^{(k)}\bigr)\Bigr)_{ij} 
                - \lambda^0_{ij}+\lambda^1_{ij}\Bigr)
    \end{align*}
\end{document}

(red lines indicate part of page layout)

As you can see, first equation protrude right text area border. This can be solved by push one term more from the first line of the first equation to the second line:

\phi_{ij}^{(k+1)}
     = {}   &  \argmax_{\phi_{ij} \in \mathbb{R}^k}
                \Bigl\langle \phi_{ij},
                \Bigl(\nabla\cdot\bigl(2u^{(k+1)}-u^{(k)}\bigr)\Bigr)_{ij}  \\
            &    - \Bigl(\nabla_G\cdot\bigl(2w^{(k+1)}-w^{(k)}\bigr)\Bigr)_{ij}
                - \lambda^0_{ij}+\lambda^1_{ij}
                \Bigr\rangle 
              - \frac{1}{2\tau}\bigl(\phi_{ij} - \phi_{ij}^{(k)}\bigr)^2    \\[2ex]

which gives:

In the case, that you may define page layout with wider text area, for example by employ geometry package, a possible solution can be:

\documentclass{article}
\usepackage{geometry}
\usepackage{mathtools, amssymb}
\DeclareMathOperator*{\argmax}{arg\,max}

\begin{document}
    \begin{align*}
\phi_{ij}^{(k+1)} 
     = {}   &  \argmax_{\phi_{ij} \in \mathbb{R}^k} 
                \Bigl\langle \phi_{ij},
                \Bigl(\nabla\cdot\bigl(2u^{(k+1)}-u^{(k)}\bigr)\Bigr)_{ij}
                - \Bigl(\nabla_G\cdot\bigl(2w^{(k+1)}-w^{(k)}\bigr)\Bigr)_{ij}
                - \lambda^0_{ij}+\lambda^1_{ij}
                \Bigr\rangle \\
            & - \frac{1}{2\tau}\bigl(\phi_{ij} - \phi_{ij}^{(k)}\bigr)^2    \\[2ex]
%
\Rightarrow 0 \overset{!}{=}{}
            & \Bigl(\nabla\cdot\bigl(2u^{(k+1)}-u^{(k)}\bigr)\Bigr)_{ij} 
                - \Bigl(\nabla_G\cdot\bigl(2w^{(k+1)}-w^{(k)}\bigr)\Bigr)_{ij} 
                - \lambda^0_{ij}+\lambda^1_{ij} 
             + \frac{1}{\tau}\bigl(\phi_{ij} - \phi^{(k)}_{ij}\bigr)       \\[2ex]
%
\phi^{(k+1)}_{ij}
\coloneqq {}&   \phi_{ij} = \phi_{ij}^{(k)} 
                + \tau \Bigl(\bigl(\nabla\cdot(2u^{(k+1)}-u^{(k)}\bigr)\Bigr)_{ij} 
                - \Bigl(\nabla_G\cdot\bigl(2w^{(k+1)}-w^{(k)}\bigr)\Bigr)_{ij} 
                - \lambda^0_{ij}+\lambda^1_{ij}\Bigr)
    \end{align*}
\end{document}

which produce:

Answer 3

Here's a solution that re-arranges your layout to (a) keep any particular row from producing an "overfull \hbox" warning, (b) reduce the sizes of the "fences" (round parentheses and square brackets), and (c) organize the material somewhat consistently the three pairs of lines.

The frame lines in the following screenshot serve to denote the left had and right hand edges of the text block.

\documentclass{article}
\usepackage{mathtools} % for '\coloneqq' macro
\usepackage{amssymb}   % for '\mathbb' macro
\DeclareMathOperator*{\argmax}{arg\,max}

\begin{document}
\begin{align*}
\phi_{ij}^{(k+1)} 
     &=  \argmax_{\phi_{ij} \in \mathbb{R}^k} \,
         \Bigl\langle \phi_{ij},
               \bigl(\nabla\cdot(2u^{(k+1)}-u^{(k)})\bigr)_{ij}
             - \bigl(\nabla_{\!G}\cdot(2w^{(k+1)}-w^{(k)})\bigr)_{ij}\\
     &\quad  - \lambda^0_{ij}+\lambda^1_{ij} \Bigr\rangle 
             - \frac{1}{2\tau}(\phi_{ij} - \phi_{ij}^{(k)})^2  \\[2\jot]
%
\Rightarrow 0 
     &\overset{!}{=}
               \bigl(\nabla\cdot(2u^{(k+1)}-u^{(k)})\bigr)_{ij} 
             - \bigl(\nabla_{\!G}\cdot(2w^{(k+1)}-w^{(k)})\bigr)_{ij} \\
     &\quad  - \lambda^0_{ij}+\lambda^1_{ij} 
             + \frac{1}{\tau}(\phi_{ij} - \phi^{(k)}_{ij}) \\[2\jot]
%
\phi^{(k+1)}_{ij} 
     &\coloneqq\phi_{ij} = \phi_{ij}^{(k)} + \tau\bigl[        
               \bigl(\nabla\cdot(2u^{(k+1)}-u^{(k)})\bigr)_{ij} 
             - \bigl(\nabla_{\!G}\cdot(2w^{(k+1)}-w^{(k)})\bigr)_{ij} \\
     &\quad  - \lambda^0_{ij}+\lambda^1_{ij} \bigr]           
\end{align*}
\end{document}