Let D1,D2 be duals of P1,P2. Both primal objectives are to minimize x 1-2x 2 with x 1,x 20. P1 has constraint -x 1+x 2=10. P2 additionally has x 1-x 2=10. Which is correct?

P2 is infeasible and D2 feasible.

D1 is infeasible and D2 feasible but unbounded.

P1 is feasible but unbounded and D1 feasible.

GATE 2020 MA – Question 18

Linear Programming · Duality · 1 mark · Multiple choice

Let D1,D2 be duals of P1,P2. Both primal objectives are to minimize $x_1-2x_2$ with $x_1,x_2\geq0$. P1 has constraint $-x_1+x_2=10$. P2 additionally has $x_1-x_2=10$. Which is correct?

Both D1 and D2 are infeasible.
P2 is infeasible and D2 feasible.
D1 is infeasible and D2 feasible but unbounded.
P1 is feasible but unbounded and D1 feasible.

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Correct answer: (A) Both D1 and D2 are infeasible.

Explanation

For P1, the dual variable y is unrestricted and must satisfy $-y\leq1$, $y\leq-2$, which require y ≥ −1 and y ≤ −2 simultaneously. D1 is infeasible. For P2, replace y by the difference of its two unrestricted dual variables; the same contradictory inequalities remain. Thus D2 is also infeasible, A. A primal and its dual can both be infeasible.

GATE 2020 MA – Question 18

Explanation

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