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Notation alert: I will use $g(i)\in \lbrace 1,2,3 \rbrace$ to denote the group into which item $i$ is placed. So your $D_i z_i$ is my $D_i x_{g(i)}.$ Introduce new continuous variables $w_i$ to ...

Why do you want derivative-free (see below)? A good derivative-based local optimization solver should be more robust and likely faster executing than a derivative-free solver. If you insist on using a ...

First of all, I'd not use terms like slave in the 21st century. For me it's the main problem & sub-problems. Most modern solvers like Gurobi can approach MINLP using decomposition methods. The ...