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.
CodePudding user response:
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.