5

You just seem to have hidden a long list of constraints of the form $(x_i=j) \Rightarrow \text{equalities}_{ij}$ Introduce a binary matrix $C_{ij}$ with $\sum_j C_{ij}= 1$ and $C_{ij} \Rightarrow \{x_{i} = j, \text{equalities}_{ij}\}$ To model the binary implication you can use big-M modelling, e.g. $-M(1- C_{ij})\leq x_{i} - j\leq M (1-C_{ij})$ and similar ...


2

This is by no means a complete list, but this is some of the best stuff for this mathematical structure: Open source: Couenne, SCIP Commercial and free: Octeract Engine (our own solver). Commercial and not free: Gurobi, Lindo Global, Baron, Antigone, LocalSolver. You might be able to get a free version of Gurobi (and maybe some of the other non-free ones) if ...


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