How can I express this max-min in CPLEX?

Question

Initially, I had the below objective function

$\max \sum_{u=1}^{U}\sum_{c=1}^{C}x_{u,c}d_{u,c}$

where $x_{u,c}$ are optimization variables

I modelled this in CPLEX as

   IloExpr  objFun (env); 
    for(int u = 0; u < U; u++){
          for(int c = 0; c < C; c++){
         objFun += x[u][c]* d[u][c];
        }
    }
    model.add(IloMaximize(env, objFun));  
    objFun.end();

Now, I have a slightly different objective

$\text{maximize} \min_{u} \sum_{c=1}^{C}x_{u,c}d_{u,c}$

Now how can I model this?

Answer 1

You can model this as a maxmin problem by introducing an auxiliary variable $\theta$:

\begin{align} \max&\quad\theta &\\ \text{s.t.}&\quad\theta \leq \sum_{c=1}^C x_{uc}d_{uc} & \forall u=1,\dots,U \end{align}

For future reference, if in contrast you had a minmax objective instead of a maxmin objective, you could apply the same trick: \begin{align} \min&\quad\theta &\\ \text{s.t.}&\quad\theta \geq \sum_{c=1}^C x_{uc}d_{uc} & \forall u=1,\dots,U \end{align}

Answer 2

Maximize an auxiliary variable $z$ subject to the constraints $z\le \sum_{c=1}^C d_{u,c}x_{u,c}\ \forall u$.