# Can we reformulate a min-max nonlinear integer programming problem with the same optimal solution?

I'm trying to solve the below min-max nonlinear integer programming problem with the following objective functions: $$\min_{x_i,y_i} \max_i \left\{\frac{x_i}{2+3y_i} + \frac{2x_i}{4+7y_i}\right\} \\ \text{subject to some linear and non-linear constraints}$$ Now, to solve this problem, I'm converting the min-max problem into a minimization problem as follows: $$\min_{x_i,y_i,z} z \\ \text{subject to } z \geq \left\{\frac{x_i}{2+3y_i} + \frac{2x_i}{4+7y_i}\right\},\forall i \\ \text{previous linear and non-linear constraints}$$ When I'm using a commercial solver to solve this reformulated problem, I'm getting a solution. However, I'm not sure if both these solutions are the same as this can be guaranteed for a linear or convex problem.

• Note that you are using the term multi-objective incorrectly. You only have one objective (which happens to involve a function called max). Multi-objective optimization is something different. Mar 8, 2022 at 17:50

Many optimization modeling systems and solvers automatically reformulate the 1st formulation as the 2nd formulation. The 2nd formulation uses the epigraph formulation of max.
It could be viewed that both of your formulations are single objective models; and to the extent there are "multiple" objective functions, they have already been crunched down to a single objective function by use of max.