@acer Thanks for your examples. Strangely, 

minimize(x^3 + y^3 + z^3 - 4*x*y*z, x = 0 .. infinity, y = 0 .. infinity) assuming z>0;

returns correct results, but the much easier version 

minimize((x - y)^2 - k*x*y, [x = 0 .. infinity, y = 0 .. infinity]) assuming (k > 0);

still returns wrong results. Any reason?

@acer For users, the following one may be better in some sense. 

eval(eval(foldl(uneval(seq), F(i, j, k), i = 1 .. Ri, j = 1 .. Rj, k = 1 .. Rk)));

Or equivalently, 

(eval@@2@foldl)(''seq'',F(i,j,k),i=1..Ri,j=1..Rj,k=1..Rk);

Since it is more direct than using _seq or S (or other dummy names), though I think that there is no essential difference for Maple between them. (But strangely, while eval(subs(...)) works, subs[eval](...) fails.)
John Fredsted's construction is simple as well, but for big (i.e., long) sequence, eval–foldl–seq version is faster in my computer.