Constrained optimization problem of 4 variables!

Question

I am stuck with this problem. I thought of trying to first solve the problem with weak inequalities for all the constraints using Kuhn Tucker conditions, and checking for solutions at which the constraints for $s$ and $t$ are slack. It is getting very messy, though. Am I on the right track or not?

Answer 1

First, forget about the strict inequalities for now -- if the maximum turns out to be on $s=0$ or $t=0$, there will simply be no solution.

You can simplify the objective function a bit by making the substitution $$s = (1+x+y)^2 \bar{s}^2$$ $$t = (1+x+y)^2 \bar{t}^2.$$ This will lead to extremely unpleasant KKT conditions which Mathematica can solve exactly to yield \begin{align*} x = y &= \frac{1}{600}\left(-251 + \sqrt{2401 + 150000\sqrt{2}}\right)\\ s = t &= \frac{\left(49 + \sqrt{2401 + 150000\sqrt{2}}\right)^2}{4500000}, \end{align*} with objective value $\approx 0.34$. Now to check the boundary cases:

1. $s\to \infty$ or $t\to \infty$: clearly the objective diverges to $-\infty$.
2. $x\to \infty$: diverges to $-\infty$ unless $s=0$, in which case the maximum is 25 at $t=2500, y=0$.
3. $y\to \infty$: clearly the same case as the previous one.
4. $x = 0$: besides the $t=0$ case already analyzed above, there is a new critical point (again, extremely unpleasant to write down) with objective value $\approx 0.33$.
5. $y=0$: same as above.

So, your problem has no maximum: the objective approaches a least upper bound of $25$ at $t=2500, y=0$, and $s\to 0$.