I have four variables $a,b,c,d$ and I want to know either $a=c$ and $b=d$ holds or $a=d$ and $b=c$ holds or both do not hold. I want to set $t=1$ if either condition holds. Else I want $t=0$.
Is there any way to check which condition holds in convex program by introducing new variables and using only real variables?
I know I can do this by checking $\max(a,b)=\max(c,d)$ and $\min(a,b)=\min(c,d)$. If both hold then I know $a=c$ and $b=d$ or $a=d$ and $b=c$ holds. However there is no natural way to take max or min operation in linear programming. Is there a way around?
Is there a way to modify the following optimization and make it work?
$n_1\leq a,b\leq m_1$ and $n_2\leq c,d\leq m_2$ and minimize $(m_1+m_2)-(n_1+n_2)$ ?