Closed-form solution to the problem: $\arg\min_{q \in \Delta_{\mathcal X} \text{ s.t } \|q-p\|_2 \le \epsilon} \mathbb E_q[X]$

Question

Let $p$ be a probability distribution on countable set $\mathcal X$, denoted $p \in \mathcal X$.

Question

• What is an "closed-form" solution for the problem: $\underset{q \in \Delta_{\mathcal X} \text{ s.t } \|q-p\|_2 \le \epsilon}{\arg\min} \mathbb E_q[X]$ ?

• What if $\mathcal X$ is finite ? Not that in that case, the problem reduces to $$ \underset{q \in \Delta_{k} \text{ s.t } \|q-p\|_2 \le \epsilon}{\arg\min} \sum_{i=1}^kq_ix_i, $$ where $k$ is the number of elements in $\mathcal X$, $x=(x_1,\ldots,x_k) \in \mathbb R^k$ (fixed), and $\Delta_k \subseteq \mathbb R^k$ is the $(k-1)$-dimensional probability simplex.