I have the following problem:
$\max_{z_{i,j,v}} \sum_{i}\sum_{j}\sum_{v} z_{i,j,v}$
$s.t. \sum_{i}\sum_{j} z_{i,j,v} = \max_{i,j} z_{i,j,v}$
$z_{i,j,v}\in[0,1]$
I can prove the constraint is quasilinear constraint, and the problem is convex peoblem(other constraint are omitted due to its obviously convexity). Yet, I can't handle the quasilinear constraint. I know it should be transformed in convex or affine terms, I don't know how to achieve it. So that's my question.
The meaning of the constraint is that only one $z_{i',j',v}$ can have non-zero value, which is also the maximum of z, the others $z_{i,j,v}, \forall i\neq i', j\neq j'$ should be zero. I am also not sure if there are better formulation of this meaning. I got an answer from cvxr.com, where a gentle man told me to introduce a binary vector to represent this constraint. I don't want to formulate this problem into a NP-complete one.
Here is my thinking. I can relax the constraint into the following inequality:
$\sum_{i}\sum_{j} z_{i,j,v} \leq \max_{i,j} z_{i,j,v}$
Note that it has to be $\textbf{less than }$or equal to, because the sum of z great than the maximum value naturaly. This inequality isn't a standard form because the -max function is concave, thus I have to deal with the -max function, the constraint can be decoupled into two constraints:
$\sum_{i}\sum_{j}z_{i,j,v} \leq t_v (1)$
$t_v \leq \max_{i,j} z_{i,j,v} (2)$
The difficulty is because the non-convexity of constraint (2), in fact, it is a $\textbf{reverse convex constraint}$. So should I keep going to deal with the reverse convex constraint? I am not sure, because the original constraint is in fact convex, after the tranformation, constraint (2) become non-convex.
So is there anyone can give some advice about this problem? Any suggestions will be grateful.
