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I just started to use Mathematica. Using the following simple command to minimize a function of variables $a,b,c,d$ with parameter $n$, it takes too much time. Do you know if there is a mistake in this command syntax?
Minimize[{((b (b - 1)/2)*(c + d) +
b*a*(c + d) + (d (d - 1)/2)*(a + b) +
c*d*(a + b))/((a + b)*(a + b - 1)*(c + d)/
2 + (c + d)*(c + d - 1)*(a + b)/2), a + c == n && b + d == n && n >= 1000 a >= 1 && b >= 0 && c >= 1 && d >= 0}, {a, b, c, d}]
Assuming[n >= 1000,
Minimize[{((b (b - 1)/2)*(c + d) +
b*a*(c + d) + (d (d - 1)/2)*(a + b) +
c*d*(a + b))/((a + b)*(a + b - 1)*(c + d)/
2 + (c + d)*(c + d - 1)*(a + b)/2),
a + c == n && b + d == n && n >= 1000 && a >= 1 && b >= 0 &&
c >= 1 && d >= 0}, {a, b, c, d}] // Simplify]
(*{n/(2 (-1 + n)), {a -> 1, b -> n, c -> -1 + n, d -> 0}}*)
n >= 1000 a >= 1orn >= 1000, a >= 1? $\endgroup$Minimize[{((b (b - 1)/2)*(c + d) + b*a*(c + d) + (d (d - 1)/2)*(a + b) + c*d*(a + b))/((a + b)*(a + b - 1)*(c + d)/ 2 + (c + d)*(c + d - 1)*(a + b)/2), a + c == n, b + d == n, n >= 1000 , a >= 1, b >= 0, c >= 1, d >= 0}, {a, b, c, d}]returns what you want. $\endgroup$