Question

Duality Theory: Consider the following LP (x1, x2 are your variables, all other values are constants):

max ax1+bx2

cx1+dx2≤e

fx1−gx2≤h

ix1+jx2≤k

x1,x2≥0

The solution to the dual has the following values (using conventional primal-dual notation in terms of variable numbering):

y1 = 0

y2 = 3

y3 = 5

With the understanding of complementary slackness, what all are constraints of the original, primal problem which we know must be tight?

1. Constraint 1

2. Constraint 2

3. Constraint 3

Answer 1

Solution :

By complimentary slackness theoram.

product of dual variable and slack value of primal constraints are 0 ....

this means If a dual variable non zero then their corresponding slack variable of primal constrains must be zero .. so that primal constraints shows equality (active constraints ) ..

given that y2 = 3 and y3 = 5 .... so, this gives us S2 = 0 and S3=0.

And other constraints 2 and 3 are shows equality.. so these two are tight

Answer = constraints 2 and 3

Let me know in the comment section if anything is not clear. I will reply ASAP!
If you liked the answer, please give a thumbs-up. This will be quite encouraging for me.Thank-you!