Let us go back to everyone's comfort zone and do some straight math this week. No probabilities, just a deterministic problem with three variables.
This cannot possibly be hard. Can it?
Think about whether this problem is actually convex or not.
Since, x.y.z >= 0, the problem is indeed convex, which means that a local minimum has to be a global one. This in turn means that we can look at the function
Now, let us look at the partial derivatives of this function and where they take the value 0:
We substitute these into the equality constraint and get:
Now, we can call 'a' the 12th root of one over lambda:
That gives us:
Alas, sixth-degree polynomials have no closed-form solution, so we would need to use a numerical solver to find that there are two real solutions, a = -1.12872 or a = 0.608340. Since a must be positive, this then gives us:
- x = 19.729758
- y = 7.301541
- z = 4.441819
The alternative? Use a solver like Seeker directly:
Which gives this solution in 2 milliseconds:
