How to model a max-min-max problem?

Everyone knows how to model max-min or min-max problems. I have a problem with objective to maximize min-max. So it can be called as a max-min-max problem. Any ideas how to model it efficiently?

The objective function looks like: $$\max\quad\min_i\max_jx_{i,j}$$ where $$x_{i, j}$$ are integer variables.

• Can you explicitly show us the objective? One possibility (perhaps not necessary?) is to formulate as a bilevel optimization problem having min-max as the inner problem, and max as the outer. Commented Mar 28, 2022 at 17:55

Introduce binary variables $$\lambda_{i,j}$$ together with the constraints $$\sum_j \lambda_{i,j} = 1 \quad \forall i. \quad(1)$$ Next, add continuous variables $$w_i$$ defined by the constraints $$w_i =\sum_{j} \lambda_{i,j} x_{i,j}.\quad(2)$$If $$x$$ is a variable, there is a trick for linearizing the products $$\lambda_{i,j} x_{i,j},$$ documented in multiple answers on this site.
Finally, introduce a variable $$z$$ together with the constraints $$z\le w_i \quad \forall i \quad(3)$$and set the objective to maximize $$z$$. $$z$$ will be the minimum of the $$w_i$$, and for any $$i$$ where $$z=w_i$$ constraints (1) to (3) plus the goal of maximizing $$z$$ will ensure that $$w_i$$ is the largest of the $$x_{i,j}.$$