# How to find an optimal solution (out of many optimal sols) of an MILP, which satisfies a condition (maximizes a variable) in GUROBI?

I have an MILP problem with an objective of the type $$\max z(x,g(x))= X - \varepsilon\cdot g(X)$$, where $$\varepsilon$$ is a penalty term determined through an iterative process. What I want is to find for a given $$z(x,g(x))$$ the optimal solution (out of all the optimal solutions) that maximizes $$X$$ (without interfering with $$\varepsilon$$). All the above to be implemented in GUROBI using python.

I'm not a Gurobi user, but what you want to look at is their "solution pool" feature. If I understand the question, you want to set the solution pool to find as many solutions as possible within a tolerance of zero (or perhaps something close to zero) of the optimal value, subject to time limits etc. This is using $$z(x,g(x))$$ as the objective function. Once Gurobi is done doing its thing, retrieve the solutions from the pool and retain only the solution(s) with maximal $$x$$ value.
Assuming your iterative process happened prior to formulating the problem, simply maximizing it with Gurobi (assuming that Gurobi can solve it to global optimality) will deterministically provide the $$X$$ point that provides the best value for your objective, since $$\epsilon$$ is not a variable in your problem.