Formulating indicator constraint set

Question

I am having difficulty formulating the indicator constraints for the following:

Consider a set of $A_{n}$ decision variables such that $A_{1},A_{2},⋯,A_{n}<A$. While all of them are integers that are either equal or greater than zero, only either one or two of them are to be non-zero, i.e., the others are to be zero. Additionally, the two non-zero variables might also be equal, though they should be unequal most of the time.

Let $A_{max}$ be the maximum value, while $A_{max2nd}$ be the second largest.

$A_{max2nd}$=$\sum\limits_{i}A_{i}-A_{max2nd}$

Let $A_{v'}$ and $A_{v}$ be two decision variables that whose values are dependant on the value of $\alpha_{i,j}$

Let $\alpha_{i,j}$ be a binary variable linking $A_{i}$ and $A_{j}$ such that

$\sum\limits_{(i,j)}\alpha_{i,j}\le1$

$\alpha_{i,j}=1$ $\implies$ $A_{v'}=A_{max}$ and $A_{v}=A_{max2nd}$ for any pairs of $i, j\in N$

if no $\alpha_{i,j}$ is equal to 1, then

$\sum\limits_{(i,j)}\alpha_{i,j}=0$ $\implies$ $A_{v}=A_{max}$ and $A_{v'}=0$ (as $A_{max2nd}$ and the other $A_{n}$s are 0, other than $A_{max}$ which must always always be present)

Appreciate your kind guidance in creating the constraints.

Thank you!

Answer 1

Do you mean $i$ and $j$ instead of $v$ and $v’$?

If so, the constraints you want are $\alpha_{i,j}\le A_i$ and $\alpha_{i,j}\le A_j$.