I have the following situation in a Mixed Integer Program: $x_1, \dots, x_n$ are binary variables, and $y, z$ are continuous. If $k$ or less variables $x_i$ are set to $1$, then I need to have $y \leq z$. That is,
$$\sum_{i=1}^nx_i \leq k \implies y \leq z$$
I need to include this conditional constraint in my MIP formulation.
One approach would be to create an auxiliary binary variable $w$ and include these big-M constraints:
$$ \sum_{i=1}^nx_i \geq k + 1 - Mw $$ $$ y \leq z + M(1 - w) $$
But, because of the structure of this condition, I have the feeling that this could be done with only one big-M constraint, without the auxiliary variable $w$. Is there another way to model that conditional constraint? If possible, it would greatly reduce the size of my formulation, because I already have lots of these constraints.
I don't see any way to avoid the extra binary variable $w$ or the two extra constraints. I do want to point out that your first constraint, while correct in spirit, is slightly off in indexing. As stated, it allows $w$ to be 0 when $\sum x_i =k$, whereas you want to force $w=1$ in that case. Changing it to $\sum x_i \ge k+1 -(k+1)w$ both fixes the issue and gives you are moderately tight choice of $M$ for that constraint.