Find $(x_1,\dots,x_n) \in (\mathbb{R}_+^*)^n$ to minimize $\sum_{k=1}^{n}x_k\prod_{i=1}^{k}{(1 + x_i)}$ such that $\sum_{k=1}^{n}{x_k} = 1$ I want to find $(x_1,\dots,x_n) \in (\mathbb{R}_+^*)^n$ to minimize
$$ \sum_{k=1}^{n}{x_k \prod_{i=1}^{k}{(1 + x_i)}} $$
with the following constraint $\displaystyle\sum_{k=1}^{n}{x_k} = 1$.

Not an answer.

We could try Lagrange multipliers. Take the Lagrangian
$$\mathcal L (\textbf{x},\lambda) = \sum_{k=1}^n x_k \prod _{j=1}^k (1+x_j) + \lambda \left (\sum _{k=1}^n x_k -1\right ) =: f(\textbf{x}) + \lambda g(\textbf{x}). $$
Find the candidate solution(s) by finding stationary point(s) i.e $\nabla\mathcal L= 0$. Get a system
$$\frac{\partial}{\partial x_i}f = (2x_i+1)\prod _{j=1}^{i-1}(1+x_j) + \sum _{k=i+1}^n \prod_{j=1 \\ j\neq i}^{i+1} (1+x_j) = -\lambda $$
with $g(\textbf{x})=0$, where for $i=1$ the first additive is simply $2x_1+1$. Unsure how to solve this analytically.
