Dual linear problems Definition. Given a standard problem (P) min c·x subject to Ax = b, x � 0 we define the dual problem (DP) as max b · y subject to A

Definition. Given a standard problem (P) min c·x subject to Ax = b, x � 0 we define the dual problem (DP) as max b · y subject to A^Ty � c.

Lemma. (Ax) · y = x · (A^Ty).

Lemma. If x ∈ Rⁿ and y ∈ R^m are feasible for (P) and (DP), respectively then b · y � c · x.

Proof.

b· y = (Ax) · y = x · (A^Ty) � x · c = c · x.

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Strong duality

Theorem 13. If x^∗ and y^∗ are optimal solutions to (P) and (DP) then b · y^∗ = c · x^∗.

Proof. We have c · x^∗ = c · x for some BFS x. Suppose that x is defined from the basis of columns B(1), . . . , B(m) etc. Define

y = (B⁻¹)^TcB.

Then y is feasible for the dual problem:

(?) A^Ty � c

(?) A_jy � cj for every j

(!) A_jy = A_j · ((B⁻¹)^Tc_B)) = (B⁻¹A_j) · c^B � cⁱ because the reduced costs are nonnegative.

Now we check that c · x = b · y:

b· y = b · ^�(B⁻¹)^Tc_B^� = (B⁻¹b) · c^B = x_B · c^B = x · c.

Corollary 14. In the above setting, if x is a basic solu- tion (not necessarily feasible) and the reduced costs are nonnegative then y is a feasible solution of the dual pro- blem.

Duality in nonstandard version

Consider (P) min c · x subject to Ax � b, x � 0. What is the dual problem?

We can change (P) to some standard problem (PS), find its dual (DPS) and read the result. This gives

Definition. The dual problem to

(P ) min c · x subject to Ax � b, x � 0 is

(DP ) max b · y subject to A^Ty � c, y � 0.

Lemma 15. If x ∈ Rⁿ and y ∈ R^m are feasible for (P) and (DP), respectively then b · y � c · x.

Proof.

b· y � (Ax) · y = x · (A^Ty) � x · c = c · x.

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Complementarity

Theorem 16. If x and y are feasible for (P) and (DP), respectively, then x and y are optimal if and only if

(a_i · x − bⁱ)y_i = 0 for i = 1, 2, . . . , m, and (A_j · y − c^j)x_j = 0 for j = 1, 2, . . . , n.

Proof. If x and y are optimal then

0 = c · x − b · y = c · x − (A^Ty) · x + y · (Ax) − b · y =

= (c − A^Ty) · x + (Ax − b) · y.