How to code this with Cplex?

Question

I have an optimization problem as below.

For $g\in G$, let $\mathcal{N}_g$ be the nodes in group $g$, and let binary variable $u_g$ indicate whether group $g$ is used. The problem has this following constraint

\begin{align} \sum_{g\in G:\ i\in N_g} u_g &= 1 &&\text{for all $i$} \end{align}

How can Model this in Cplex?

For example, G=10;

$\mathcal{N}_1={1,2}$

$\mathcal{N}_2={1,3,4}$

$\mathcal{N}_3={1,3,5}$

$\mathcal{N}_4={1,4,5,6}$

...

$\mathcal{N}_{10}={1,6,7,8}$

The only constraint says that every node must be chosen just once.

Here is what I tried so far

IloEnv env;
try{

IloNum G=10;

IloNumVarArray Ug(env, G,0,1,ILOINT)

IloModel model(env)

In order to make it efficient for implementation I do a reformulation. I generate a binary matrix,$B$ of size $N_{node}\times G$, where $N_{node}$ is the number of nodes. If node $n, n\in{1,2,\cdots,N_{node}}$ is present in group $g, g\in{1,2,\cdots,G}$, then $B_{n,g}=1$, otherwise 0.

IloExpr constFun(env)

Note: In Matlab, this constraint now can be expressed as

  for n=1:N
      
      sum(Ug(find(B(n,:)==1)))==1;
      
  end

Answer 1

In OPL CPLEX you could start with

int g=10;
range G=1..g;
range I=1..5;

{int} Ng[G]=[{1,2},{3,4},{5},{},{},
{},{},{},{},{}];

dvar boolean u[G];

subject to
{
  forall(i in I) sum(g in G:i in Ng[g]) u[g]==1;
}