How is the feasibility checked in this step of Rosen's gradient method?

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I know that feasibility can be checked by checking the value of the new objective function with the previous one and by taking the gradient ans direction. But I cannot understand what is used to find the value of α1 = 1/4 here. I'm fairly new to this topic and can't seem to understand most of the topic due to this. enter image description here enter image description here

linear-algebra math optimization algorithm
1 Answers

Note the first constraint: it requires that x1 - 2 x2 <= 0.

Now, within that Max we are looking at all possible values of alpha that are greater than zero, and we want the largest alpha value such that

1 + 4 alpha
    1

"is feasible", ie satisfies all constraints. In particular, in order to satisfy that first constraint, we must have

x1 <= 2 x2

which here is

1 + 4 alpha <= 2
    4 alpha <= 1
      alpha <= 1/4

It is clear then that the Max alpha that satisfies the first constraint is 1/4. We can check that with this alpha the other constraints are also satisfied; and we're done.

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