I have a mixed integer optimization problem that has linear objective function and two constraints, one is linear but one is not. The non-linear constraint has the form
$$x_{kn}f(\mathbf{y})\geqslant x_{kn}\sum_{j=1}^Kx_{jn},$$ for all $k,n$ in $\{1,\ldots,K\}$ and $\{1,\ldots,N\}$.
Where $x_{kn}$ are Booleans and $\mathbf{y}$ is a vector in $[0,1]^N$. The only way I have $x_{kn}$ in both sides of the above inequality is to say that the constraints is active only when $x_{kn}=1$.
Assuming that $f(\mathbf{y})$ is convex or concave, can I solve this problem efficiently through convex optimization?
I can remove the $x_{kn}$ from the right hand side of the above inequaluty by introducing a large number $M$ but that would keep the product $x_{kn}f(\mathbf{y})$.
Suppose that $f(y)$ has a known lowerbound $L$. Note that $\sum_{j=1}^K x_{jn}$ is bounded above by $K$. You can also deactivate the constraint by writing it as: $$f(y) \geq \sum_{j=1}^K x_{jn} + x_{kn}(L-K)$$ This constraint is convex when $f(y)$ is concave. When $x_{kn}=0$ you obtain the constraint $f(y) \geq \sum_{j=1}^K x_{jn}$, whereas when $x_{kn}=1$, the right hand side is at most $L$, so the constraint does not put any restriction on $y$ or $x_{kn}$.