maximize the maximum value of a linear optimization

Question

Here is my problem, first, $v$ is the optimal value of a LP problem $$ v = \max_w w^TF\beta $$ subject to $$ w \ge 0 \\ Aw = b \\ $$

Then I want to find the coefficient $\beta$ that maximize $v$ $$ \beta^* = argmax_{\beta} \ \ v $$