I have a MILP model which defines a relation between three real variables $x$, $y$ and $s$ where $s = f(x,y)$.

I want to optimize the following objective: $$\min_x\max_y s$$

How can I achieve it?

The problem is MILP because the relation $f$ stacks linear operations between $x$ and $y$ with non-linear operations that rely on binary variables.
To be more specific, $f$ is a simple deep neural network with ReLU activations encoded with indicator contraints.

  • 1
    $\begingroup$ You said "MILP" (so presumably $f()$ is linear), but specified that $x,y,s$ are all real-valued. Are there other (integer) variables? Can you give us a reasonably compact representation of the full problem? $\endgroup$
    – prubin
    Commented Sep 7, 2022 at 16:19
  • $\begingroup$ @prubin I edited the question, ty $\endgroup$
    – Daniele
    Commented Sep 7, 2022 at 16:48
  • $\begingroup$ Input Convex Neural Networks? $\endgroup$
    – xd y
    Commented Sep 9, 2022 at 9:59
  • $\begingroup$ Still not enough information on this. You don't have any constraints? $\endgroup$
    – Brannon
    Commented Oct 5, 2022 at 2:56
  • $\begingroup$ @Brannon i do have several indicator constraints used to encode the neural network $f()$ $\endgroup$
    – Daniele
    Commented Oct 8, 2022 at 11:42


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