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I got bad news for you: The problem is as long as you care about the inner max being the actual max and not just some local maxima (which might make sense with multiple roll outs for a game theory simulation) you are stuck with calling an global optimizer to just evaluate $x \mapsto \max_{y_\in Y} f(x,y)$ correctly. If $\dim(y)$ is small you might want to ...

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This answer complements that by @worldsmithhelper . Not only does an objective function not have to include all the decision variables, it need not include any of them. An "optimization" problem having a constant (zero) objective function, which is equivalent to not having an objective function, is called a feasibility problem. Your understanding ...

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NO Not all decision variables need to be present in the objective. They can be often used to handle constraints or express other things. Consider this simple problem: $$\min_{x,y_1,y_2,y_3,y_4,y_5 \in \mathbb{N}} x \text{ subject to }$$ $$\forall i \in \{1,2,3,4,5\}: \ 0 \leq y_i \leq x$$ $$y_1+y_2+y_3+y_4+y_5 = 12$$ Which might answer the question how much ...

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Disclaimer: One might want to look for a reformulation or a special structure to apply mathematical tools to find optimal in the feasible set. I am assuming you're already past the possibility that your problem case could be reformulated as a MILP/LP/QP etc. So, with the problem on hand, we're dealing a case where we cannot have a reformulation. Whatever I ...

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Julia Niebling and Gabriele Eichfelder recently published a paper on solving nonconvex multiobjective optimization problems using branch and bound. You can find the paper through the following link https://doi.org/10.1137/18M1169680

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Yes we can. Branch and bound can both deal the integer variables and with the potenial non convexities of the non-linear functions. Most branch and bound methods can also handle constraints. Depending on the solver you might need provide an relaxation for the bounding part. However multi objective branch and bound methods are harder to come by, i am still ...

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One traditional method to linearize multi-linear/polynomial constraints are McCormick envelopes, which are refined over time. Here is a great resource on those. A single objective solver that uses them and improves them in an interesting way is Alpine to get an quick idea whether you could adapt ideas from it for you i would recommend this 22 minute ...

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