The following unusual optimization problem came up and I don't know where to begin:

Maximize over the real variables $x_1, \dots, x_n$ the sum $$ S = \sum_{r = 1}^n \frac{1}{x_1 + \dots + x_r} $$ subject to the constraints $1 \leq x_1 \leq x_2 \leq \dots \leq x_n$ and subject to $$ p^{x_1} + \dots + p^{x_n} \leq t p $$

Here $p < 1$ is a constant which may become very small, and $1 \leq t \leq n$.

Any upper bound on $S$ would be useful. Clearly $S \leq \log n$, but for $t$ small much better constraints should be available.

If you use the Lagrange multiplier approach to constrained optimization you get equations of the form $$ p^{x_{i+1}} = p^{x_i} - \frac{\lambda}{(x_1 + \dots + x_i)^2} $$ for some $\lambda > 0$. (This ignores the boundary constraint $x \geq 0$)

This gives a recursive equation to solve for all $\hat x_i$ in terms of $\hat x_1, \lambda$. Experiments show that the sequence $\hat x_i$ increasingly converges to some constant $y$, and this convergence is rapid.

If you could show that the convergence was rapid enough, then you would basically know that critical situation is one in which all the $x_i$ are equal. It is quite easy to bound $S$ in this case.