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The code is working fine but I feel as my intuition could be wrong.

I first set up my problem as: $$\min_{f(x)\leq F} g(x) $$ and gurobi was unable to find a feasible solution. I then relaxed optimality and only asked for $g(x)$ to below some threshold $G$ and reformulated the problem as: $$\min_{f(x)\leq F, g(x)\leq G} 0 $$ and was able to obtain a solution somewhat quickly. I could even sequentially lower $G$ to find the optimal solution.

Is there a deep reasoning why the first approach cannot find a feasible solution but the second had no issue?

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    $\begingroup$ A mixed-integer optimizer is a complicated beast. Maybe, the objective cut in the second formulation for some reason helps presolve to reduce the problem dramatically. $\endgroup$ Commented May 24 at 4:43

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