Linear Programming Lecture Notes - Penn State Personal Web Server

3. Strong Duality
Lemma 9.7. Problem D has an optimal solution w∗ ∈ Rm if and only if there exists
vector x∗ ∈ Rn and s∗ ∈ Rm such that:
∗
w A ≥ c
Primal Feasibility
w ∗ (9.15)
≥ 0
⎧
⎪⎨
Ax
Dual Feasibility
⎪⎩
∗ + s ∗ = b
x ∗ ≥ 0
s ∗ (9.16)
≥ 0
∗ ∗
(w A − c) x = 0
(9.17) Complementary Slackness
w ∗ s ∗ = 0
Furthermore, these KKT conditions are equivalent to the KKT conditions for the primal
problem.
Proof. Following the proof of Lemma 9.2, let β = −bT , G = −AT , u = wT and
κ = −cT . Then the dual problem can be rewritten as:
⎧
⎪⎨
max βu
s.t. Gu ≤ κ
⎪⎩
u ≥ 0
Let x T ∈ R 1×n and s T ∈ R 1×m be the dual variables for this problem. Then applying
Theorem 8.7, we obtain KKT conditions for this problem:
(9.18)
(9.19)
(9.20)
We can rewrite:
∗
Gu ≤ κ
Primal Feasibility
u ∗ ≥ 0
⎧
⎪⎨
x
Dual Feasibility
⎪⎩
∗T G − s ∗T = β
x ∗T ≥ 0
s ∗T ≥ 0
∗T ∗
x (Gu − κ) = 0
Complementary Slackness
Gu ∗ ≤ κ ≡ −A T w T ≤ −c T ≡ wA ≥ c
s ∗T u ∗ = 0
x ∗T G − s ∗T = β ≡ x ∗T (−A T ) − s ∗T = −b T ≡ Ax ∗ + s ∗ = b
x ∗T (Gu ∗ − κ) = 0 ≡ x ∗T (−A T )w ∗T − (−c T ) = 0 ≡ (w ∗ A − c)x ∗ = 0
s ∗T u ∗ = 0 ≡ s ∗T w ∗T = 0 ≡ w ∗ s ∗ = 0
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