4 5 points Consider a mathematical program with the followin

4) (5 points) Consider a mathematical program with the following constraints: x1 - 2x2 + 3x3 = 0 Determine the largest step size possible (could be infinity) that preserves feasibility in the direction Delta x = (-3, -3, 9) at the point x = (9, 4, 6). Does the step side indicate that the model is unbounded?

Solution

The constraints could be rewritten as:

x₁-2x₂+3x₃+x₄ = 25

x₁,x₂,x₃,x₄ >=0

Let the initially x₄ =0.Now let us increase the value of x₄ by t

_Then x₁ = 25 + 2x₂-3x₃-x₄ = 25+2(4)-3(6)+t = 25+8-12+t = 21+t

Now for preserving feasibility, x₁>0 => 21+t>=0 => t>=-21

So the minimum value of t that we can take is -21 and largest value is infinity

hence the new solution with this becomes: (infinity, 4,6)

Yes t indicates that the model is unbounded