Consider the LP
$min f(x_1,...,x_n) = \sum_{i=1}^n i x_i $
such that
\begin{align*} x_1 \geq 1 \\ x_1 + x_2 \geq 2 \\ ... \\ x_1 + ... + x_n \geq n \end{align*}
Im trying to find the Dual and an optimal solution to the LP
The dual is easy to find:
$max f = \sum_{i=1}^n i x_i $
such that
\begin{align*} x_1+x_2+...+x_n \geq 1 \\ x_2 + ... + x_n \geq 2 \\ ... \\ x_n \geq n \end{align*}
but, how can we find the optimal solution?
The dual is
$$\max_y \sum_{i=1}^n i y_i$$
subject to $$y_i \ge 0$$
$$\sum_{i=j}^n y_i = j, \forall j \in \{1,\ldots, n\}$$
In particular, for $n>1$, we have $$y_1 +\ldots + y_n =1$$
and $$y_2 +\ldots + y_n =2$$
which implies that $y_1=-1<0$ which shows that the dual is infeasible.
We can check that the primal is feasible and hence it is unbounded.
To directly see that the primal is unbounded for $n>1$, notice that $(1+k, 1, \ldots, 1, 1-k)$ is feasible for the primal. We can let $k$ be arbitrary big and the objective function will be unbounded.
