Optimal probability mass function with domain $\mathbb N^+$ Given nonnegative infinite sequence $\left( y_k \right)_{k \geq 1}$ with $\displaystyle\sum_{k \geq 1} y_k = 1$, solve the optimization problem
$$\begin{array}{ll} \text{maximize} & \displaystyle\sum_{k \geq 1} k \, x_k \, y_k\\ \text{subject to} & \displaystyle\sum_{k \geq 1} k \, x_k = a\\ & \displaystyle\sum_{k \geq 1} x_k = 1\\ & x_k \geq 0\end{array}$$
What about the infinite sequence $\left( x_k \right)_{k \geq 1}$ that minimizes the sum?
I made this problem up.  I am interpreting $\left( x_k \right)_{k \geq 1}$ and $\left( y_k \right)_{k \geq 1}$ as probability distributions over the positive integers and $a$ is an expected value. My guess is this type of problem has been well studied, which is why I am asking here. Thank you.
Edit:
After doing some research, it appears that the above problem may have a solution which can be computed with linear programming.  However, there is little to no chance of there being a nice form for the solution if the $y_n's$ are arbitrary.
So to make the problem more manageable, I posted a modified problem:
Optimization of Probability Mass Function Linear Programming

I don't have a complete answer, but I'll go ahead and share my thoughts...
We cannot have $a < 1$, because this implies the problem is infeasible.
If $a = 1$, the only feasible solution is $x_1 = 1$.
If $a > 1$, let $j \equiv \arg \max_k k y_k$ (assuming there is a maximum). There are then 3 sub-cases to consider:


*

*If $j = a$, set $x_j = 1$.  You are done.

*If $j < a$, the optimal value is not achieved, but is approached by a sequence of solutions of the form $x_j = 1 - \delta$, $x_n = \delta$, where $\delta$ satisfies $n \delta + j(1-\delta)=a$.  As $n$ increases, $\delta$ decreases, and the objective value approaches $j y_j$.

*If $j > a$... I don't know.

