Linear Programming Lecture Notes - Penn State Personal Web Server

The (negative) identity rows in E ensure that ˆx ≥ 0. It follows that:
cˆx = c T (x ∗ + λd) = cx ∗ + λcd > cx ∗
That is, this point is both feasible and has a larger objective function value than x∗ .
We can now apply Corollary 8.5 to show that there are vectors w and v so that:
w v
AE
(8.29)
= c
E
(8.30)
(8.31)
w ≥ 0
v ≥ 0
Let I be the indices of the rows of A making up AE and let J be the indices of the
variables that are zero (i.e., binding in x ≥ 0). Then we can re-write Equation 8.29 as:
(8.32)
wiAi· −
vjej = c
i∈I
j∈J
The vector w has dimension equal to the number of binding constraints of the form Ai·x = b
while the vector v has dimension equal to the number of binding constraints of the form
x ≥ 0. We can extend w to w ∗ by adding 0 elements for the constraints where Ai·x < bi.
Similarly we can extend v to v ∗ by adding 0 elements for the constraints where xj > 0. The
result is that:
(8.33)
(8.34)
w ∗ (Ax ∗ − b) = 0
v ∗ x ∗ = 0
In doing this we maintain w ∗ , v ∗ ≥ 0 and simultaneously guarantee that w ∗ A − v ∗ = c.
To prove the converse we assume that the KKT conditions hold and we are given vectors
x ∗ , w ∗ and v ∗ . We will show that x ∗ solves the linear programming problem. By dual
feasibility, we know that Equation 8.32 holds with w ≥ 0 and v ≥ 0 defined as before and
the given point x ∗ . Let x be an alternative point. We can multiply both side of Equation
8.32 by (x∗ − x). This leads to:
(8.35) wiAi·x ∗ −
vjejx ∗
− wiAi·x −
vjejx ∗
= cx ∗ − cx
i∈I
j∈J
i∈I
We know that Ai·x ∗ = bi for i ∈ I and that x ∗ j = 0 for j ∈ J. We can use this to simplify
Equation 8.35:
(8.36)
wi (bi − Ai·x) +
vjejx = cx ∗ − cx
i∈I
j∈J
The left hand side must be non-negative, since w ≥ 0 and v ≥ 0, bi − Ai·x ≥ 0 for all i,
and x ≥ 0 and thus it follows that x ∗ must be an optimal point since cx ∗ − cx ≥ 0. This
completes the proof.
127
j∈J