Minimization problem with infinitely many variables and linear constraints How can the following minimization problem be solved?
$$\begin{array}{ll} \text{minimize} & \displaystyle\sum_{i=1}^{\infty}P_i^3\\ \text{subject to} & \displaystyle\sum_{i=1}^{\infty} P_i = 1\\ & P_i \geqslant 0 \quad\text{for} \quad i \in \mathbb N^+\end{array}$$
I guess the Lagrange multipliers and Karush–Kuhn–Tucker conditions won't work for infinitely many variables. Any hints on how to approach the problem will really be appreciated.

Generally, a major part of such problems is establishing that
a solution exists.
The problem doesn't have a solution in the usual sense.
However, $\inf \{ \sum_k p_k^3 |  \sum_k p_k =1, p_k \ge 0 \} = 0$.
To see this, note that the cost always non negative, and taking $p_1=\cdots = p_n = {1 \over n}$ and $p_k = 0$ for $k >n$, we have
$\sum_k p_k^3 = {1 \over n^2}$. Hence the $\inf$ is zero.
