In any case, I'm scratching my head. For example:
> We can thus, for any Brainfuck program, create an integer optimization problem with 3n decision variables where the optimal solution is that program.
Integer optimization is designed to solve numerical problems. Is there any reason or inspiration to suggest that use this technique is a productive way to write programs that satisfy certain constraints?
You can definitely express any Brainfuck program as a integer program, and you can add constraints that will constrain what the program does/must do, but it isn't clear to me how to add useful constraints/objective that are still linear. It'd be an interesting exercise though, and something I'll be thinking about.
Integer optimization does have some practical uses related to programming though - its use has been explored in compilers. E.g. http://www.sciencedirect.com/science/article/pii/S0898122197...
AMPL is one such language