[Tex/LaTex] How to break a long equation with multi-alignment (two-column)

Question

This question was asked before:

1. How to break a long equation?
2. How can I split an equation over two lines

But what I want to ask is a bit complicated than both.

\begin{equation}\label{eq:dual_algo}
\begin{aligned}
Z_{t+1}^i &= \underset{Z^i\succeq 0, \operatorname{tr(Z^i)=1}}{\text {argmax}}
 & & \langle \mathcal{A}_d^*(D^i)-\sum_{j:(i,j)\in \mathcal{E}} [Y_t^{ij}
-\alpha X_t^j],Z^i \rangle \\ & & &- \frac{|\mathcal{N}_i\alpha|}{2}\|Z^i\|_F^2\\ 
X_{t+1}^i &= \underset{X^i\succeq 0, \operatorname{tr(X^i)=1}}{\text{argmax}} 
& &\langle \sum_{i:(i,j)\in \mathcal{E}} [Y_t^{ij}+\alpha X_t^j],X^j \rangle \\
& & &- \frac{|\mathcal{N}_j\alpha|}{2}\|X^j\|_F^2
 \\
Y_{t+1}^{ij} &= Y_{t}^{ij} - \alpha (Z_{t+1}^j -Z_{t+1}^i)
\end{aligned} 
\end{equation}   

The following is what I get:

But I want the following:

Therefore, there are a few requirements:
1. Equations cannot exceed the border
2. Three = should be aligned
3. Right-hand-side description should be aligned (the first two are too far away from "argmax")
4. The fractional part should also be aligned.

It is double column.

How do I fix it?

Answer 1

Hope this will be helps:

\begin{equation}\label{eq:dual_algo}
\begin{aligned}
Z_{t+1}^i &= \underset{Z^i\succeq 0, \operatorname{tr(Z^i)=1}}{\text {argmax}}  \langle \mathcal{A}_d^*(D^i)-\sum_{j:(i,j)\in \mathcal{E}} [Y_t^{ij} -\alpha X_t^j],Z^i \rangle \\
 & \quad  - \frac{|\mathcal{N}_i\alpha|}{2}\|Z^i\|_F^2\\
X_{t+1}^i &= \underset{X^i\succeq 0, \operatorname{tr(X^i)=1}}{\text{argmax}} \langle \sum_{i:(i,j)\in \mathcal{E}} [Y_t^{ij}+\alpha X_t^j],X^j \rangle \\
& \quad - \frac{|\mathcal{N}_j\alpha|}{2}\|X^j\|_F^2 \\
Y_{t+1}^{ij} &= Y_{t}^{ij} - \alpha (Z_{t+1}^j -Z_{t+1}^i) 
\end{aligned}
\end{equation}

To align the description give \quad or \qquad.