1 Consider an LP in standard form which has as its first con

1. Consider an LP in standard form which has as its first constraint

x1 +x2 x3 x4 1.
Assume that you know (based on other constraints) that x2 = 0.

Show that the dual variable y1 associated with the first constraint is zero.

Solution

Assume that you know (based on other constraints) thatx₂= 0. Show that the dual variabley1associated with the first constraint is zero. We immediately deduce that -x₁+x₂-x₃-x₄= -x₁-x₃-x₄. Now making the assumption that all variables are positive (true for LP\'s in standard inequality form) we have-x₁-x₃-x₄<0<1 and hence the slack variable of this constraint is at least 1. Now, by the Theorem of Complementary Slackness, we deduce that y₁= 0.