min_max and minimize

Dear All,

I have been trying to find the fundamental differences between min_max and minimize predicates in fd library. Apart from the basic definitions of the two predicates and the fact that for example, minimize(?Goal, ?C)  does not take negative C, that is, -C, are there any other operational differences between the two? I tried to understand them by using both on the same minimization problem, I found out that even though they find the same optimum value, they took different computing times to find the provable optimum value. I will be very grateful with some explanations on the working pattern of the min_max/2 when used to solve maximization problems, that is, when it is set up as min_max(?Goal, ?-C). Any information will be highly aprreciated.

thanking you,

Farouk