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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 ...