Thank you for your answers. Here is the complete version of the problem with constraints:
sum(sum( x(r,t) * w(r) * e(r) * f(r)² ))
such that : 0<= x(r,t) <=1 ; 0<= f(r) <=f
sum( x( r, t1:t2) ) =0 for every r
sum( x( r, t2+1:t3) ) <=1 for every r
sum( x( r, t3+1:T) ) = 0 for every r
sum( x( r,t) * f(r)) = 0 for every t
R * (1-epsilon) *( sum (x(r,t2+1:t3))^-1 <=1
x(r,t) * w(r)* e(r)* f(r)^-1 *To^-1 <= 1
Variables of the program: x(r,t) and f®.
Now , in the beginning my problem was a mixed integer non linear program (I know thre are solvers dedicated for this kind of problems) . After I relaxed it by imposing that x(r,t) belongs to [0,1].
To resolve it I tried to write it using the geometric programming form you propose. Except that for it to work, all my variables should be strictly positive as I understood since it will be considering the log of the objective function. Is it normal if i get as optimal solution zeroes for both x(r,t) and f® ?
Thank you correcting me if I’m on the wrong path.