Hey Henrik,
that does help, thank you! I'll have to look into interpolated functions in Mathematica sometime.
I actually got it to work by using Minimize instead of NMinimize and simplifying the definitions of beta and gamma by reusing alpha. I was actually interested in a slightly different problem (with another piecewise function on the RHS instead of just 1/3 and some conditions to remove the trivial w=x=y=z=1 solution), but that change didn't pose any problems.
Kind regards,
Jesco