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}
![enter image description here](https://webcf.waybackmachine.org/web/20210929175844im_/https://i.stack.imgur.com/8Lvc1.png)
(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:
![enter image description here](https://webcf.waybackmachine.org/web/20210929175844im_/https://i.stack.imgur.com/PA8qS.png)
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:
![enter image description here](https://webcf.waybackmachine.org/web/20210929175844im_/https://i.stack.imgur.com/GjDCJ.png)
&
, which is the designated indicator for the alignment point. You seem to be aware of this because you have mentioned&
in the title. The&
is best placed before a sign of relation, and if a continuation line does not include a sign of relation, place the&
at the beginning followed by\quad
. If you omit the&
from any line, that line will be assumed to be wanted on the left-hand side, shoving all other lines to the right.