I came across this Theorem on Optimum solution to a Linear programming problem:
" If $S$ is the feasible region of some linear program with objective function $ z=c^{T}\textbf{x}$ then 1) $z$ attains its optimal value at some vertex of $S$, 2) the linear program is infeasible, or 3) the linear program is unbounded. "
But, if we consider $\text{max}_{\textbf{x}\in \mathbb{R}^2}$ $x_1+x_2$ s.t. $x_1+x_2=1$ and $\textbf{x} \geq \textbf{0},$ then it is clear that $(0.5,0.5)$ is an optimal solution but it does not lie on vertex of the feasible region (which is a polyhedra).
Hence, I think (1) means that if $X$ is an extreme point of $S,$ then $X$ corresponds to an optimal solution(but converse is not necessarily true) ?
Could anyone advise please? Thank you.