Aligning two-level optimization problem

Question

I am trying nicely (and correctly) align two level optimization problem. At the moment came up with the following solution:

$
\begin{array}{llr}
    \min \limits_{\mathbf{x}, \mathbf{y}} & -2x_1^2 + x_2^2 - 6y_1 + y_2^2 \\
    \st & x_1^3 + 3x_2 - 10 \le 0 \\
        & \displaystyle \mathbf{y} \in \argmin_{\mathbf{y}\in [0, 10]^2} \; 2x_1^2 + 2y_1^2 - 8y_2 \\
            & \hspace{1.25cm} \st - x_1 + 2x_1 - x_2^2 + 2y_1 - y_2 - 3 \le 0\\
            & \hspace{1.9cm}     - x_1 + 2x_1 - x_2^2 + 2y_1 - y_2 - 3 \le 0 \\
      & \mathbf{x} \in [-10, 10]^2,\; \mathbf{y} \in [-10, 10]^2
\end{array}
$

Which produces the following output:

Here I highlighted couple issues:

1. What would be a better way (compared to manual hspace{} the first constraint starting with keywords "s.t." align based on the upper line min keyword (see blue line in the picture). The rest constraints should be left-aligned by the first constraint.
2. Why am I getting different spacing for the inner constraints? Both are completely the same, but in the first constraint there is a bigger space after the minus sign (situation highlighted in picture in red "circle")

Answer 1

Instead of the array environment, I'd suggest to try aligned (which is basically an array but with spacing more suitable for math). The MWE could be the following:

\documentclass{article}
\usepackage{amsmath}
\DeclareMathOperator{\st}{s.t.}
\DeclareMathOperator*{\argmin}{arg\,min}
\begin{document}

\[
\begin{aligned}
    &\min \limits_{\mathbf{x}, \mathbf{y}} && {-}2x_1^2 + x_2^2 - 6y_1 + y_2^2 \\
    &\st && x_1^3 + 3x_2 - 10 \le 0 \\
        &&&\mathbf{y} \in
     \begin{aligned}[t]
        &\argmin_{\mathbf{y}\in [0, 10]^2} && 2x_1^2 + 2y_1^2 - 8y_2 \\
            &\st &&{-}x_1 + 2x_1 - x_2^2 + 2y_1 - y_2 - 3 \le 0\\
            &&&{-}x_1 + 2x_1 - x_2^2 + 2y_1 - y_2 - 3 \le 0
     \end{aligned}\\
      &&&\mathbf{x} \in [-10, 10]^2,\; \mathbf{y} \in [-10, 10]^2
\end{aligned}
\]
\end{document}

Note that I defined \st as a math operator. It's just to get spacing around it similar to \max and \argmax, otherwise the correct alignment would be trickier to produce. Another idea is to typeset the inner optimization problem using nested aligned, which is cleaner. But don't forget to brace the unary minuses, as aligned automatically thinks that an operator following & is binary.