I have a constraint as follows:

$ \sum_i {r_i} \geq \max \{g_j, B_j\} $

where, $r_i$, $g_j$ are variables and $B_j$ is a parameter.

  1. How do I linearize the constraint (I suppose using big-M method)?

  2. Since, I have a lot of elements under the set $i$, I would like to have the option of big-M. However, another option I was thinking is to use the following two constraints:

$\sum_i {r_i} \geq g_j$

$\sum_i {r_i} \geq B_j$

Therefore, the bigger of the $g_j$ and $B_j$ will be selected for the minimization problem and the other will be redundant. For example, for a minimization problem if we have $x \geq 2$ and $x \geq 5$, then $x \geq 2$ will be a redundant constraint.

  • $\begingroup$ Your original question had $\le \max$. For $\ge \max$ instead, you do not need a binary variable, and your proposed formulation is correct. $\endgroup$
    – RobPratt
    Commented Jun 8, 2020 at 21:10