We want to minimize the following function:
\begin{align}
&\min_{x_1,y_1,x_2,y_2} \underbrace{\frac{x_1^2+y_1^2+x_2^2+y_2^2}{(x_1y_2-x_2y_1)^2}}_{f}+(\alpha-1)\underbrace{\frac{(x_1+\beta y_1)^2+(x_2+\beta y_2)^2}{(x_1y_2-x_2y_1)^2}}_{g}, \label{p22}\\ \\
&s.t. \hspace{2mm} x_1,y_1,x_2,y_2 \in \{-\frac{M}{2},\ldots,0, \ldots, \frac{M}{2}\},\\
&\hspace{6mm} (x_1,y_1)\neq(ax_2,ay_2)\hspace{3mm} for\hspace{3mm} a\in \mathbb{R} , \label{line}\\ %\setminus \{0\}
&\hspace{6mm} (x_1,y_1)\neq(0,0), \hspace{3mm} (x_2,y_2)\neq(0,0) , \label{line2}
\end{align}
In the following, $\alpha$, $\beta$ and $M$ are given. We already know what values of $x_1,y_1,x_2,y_2$ will separately optimize $f$ and $g$. In other words, we know the solution when each function is minimized, separately. Now, we want to optimize their summation, when $g$ is multiplied by $(\alpha-1)$. We know that optimal variables for minimizing only the $f$ are $\{x_1,y_1,x_2,y_2\}=\{-\frac{M}{2}, \frac{M}{2},\frac{M}{2}, \frac{M}{2}\}$. Also, we know that optimal variables for minimizing only the $f$ are $\{x_1,y_1,x_2,y_2\}=\{-\frac{M}{2}, \frac{M}{2},\frac{M(1-2\beta)}{2}, \frac{M}{2}\}$. The point is that three variables have exact same values in both individual optimization and the only difference is $x_2$. If we can say that in the final solution of our optimization problem ($f+(\alpha-1)g$), the other three variables ($x_1,y_1,y_2$) will take the same values (i.e. $f$ and $g$ are not competing in $x_1$, $y_1$ and $y_2$ fields), the value of $x_2$ to solve our optimization problem ($f+(\alpha-1)g$) will be simply acquired (we will relax the discrete constraint on it, take the derivative and find optimal $x_2$). However, one can make the following argument about the assumption of other three variables ($x_1,y_1,y_2$). One can say:
The optimal values of $\{x_1,y_1,y_2\}$ for $f$ are equal to $\{-\frac{M}{2}, \frac{M}{2}, \frac{M}{2}\}$ only when $x_2=-\frac{M}{2}$, and if $x_2$ is changed, in general, change of $x_2$ may change the optimal values of other three variables. The same holds for $g$, as well. So, you cannot say that in the optimization of $f+(\alpha-1)g$, the values of $\{x_1,y_1,y_2\}$ are same the case of the individual optimization of $f$ and $g$, even if they are not competing in those fields.
We know that this claim of is not true in our case, by using the global search, but this is not a mathematical proof, and so cannot be used to answer the argue. Now, my question is that can one raise this issue? If yes, how should answer that in this case this problem is not of concern?