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.

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    $\begingroup$ 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. $\endgroup$ Commented Mar 28, 2022 at 17:55

1 Answer 1


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 $x_{i,j}.$


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