CSC 445/545 Notes 2.4: Primal and Dual Feasible Solutions

The Primal Problem:

Maximize C1*X1  + C2*X2 + .... + Cn*Xn 

subject to

A11* X1  +  A12*X2  +  ...  + A1n*Xn   <= B1
A21* X1  +  A22*X2  +  ...  + A2n*Xn   <= B2

Am1* X1  +  Am2*X2  +  ...  + Amn*Xn   <= Bm

X1, X2, ... , Xn >= 0

The Dual Problem

Minimize B1*Y1  + B2*Y2 + .... + Bm*Ym 

subject to

A11* Y1  +  A21*Y2  +  ...  + Am1*Ym   >= C1
A12* Y1  +  A22*Y2  +  ...  + Am2*Ym   >= C2

A1n* Y1  +  A2n*Y2  +  ...  + Amn*Ym   >= Cn

Y1, Y2, ... , Ym >= 0

Theorem

For every primal feasible solution
x= (X1, X2, ... , Xn) and
for every dual feasible solution
y= (Y1, Y2, ... , Ym)
C1*X1 + C2*X2 + .... + Cn*Xn <=
B1*Y1 + B2*Y2 + .... + Bm*Ym

Proof

Because the constraints of the dual are that

A11* Y1  +  A21*Y2  +  ...  + Am1*Ym   >= C1
A12* Y1  +  A22*Y2  +  ...  + Am2*Ym   >= C2

A1n* Y1  +  A2n*Y2  +  ...  + Amn*Ym   >= Cn

we have that S= 

(A11* Y1  +  A21*Y2  +  ...  + Am1*Ym)*X1 +
(A12* Y1  +  A22*Y2  +  ...  + Am2*Ym)*X2 +

(A1n* Y1  +  A2n*Y2  +  ...  + Amn*Ym)*Xn

>= C1*X1  + C2*X2 + .... + Cn*Xn .

Regroup the terms on the left hand side so that they are grouped according the Yi instead of Xi:


S= 

(A11* X1  +  A12*X2  +  ...  + A1n*Xn)*Y1 +
(A21* X1  +  A22*X2  +  ...  + A2n*Xn)*Y2 +

(Am1* X1  +  Am2*X2  +  ...  + Amn*Xn)*Ym 

From the primal problem, 

A11* X1  +  A12*X2  +  ...  + A1n*Xn   <= B1
A21* X1  +  A22*X2  +  ...  + A2n*Xn   <= B2

Am1* X1  +  Am2*X2  +  ...  + Amn*Xn   <= Bm

So, S <= B1*Y1  + B2*Y2 + .... + Bm*Ym 

So in conclusion:
C1*X1  + C2*X2 + .... + Cn*Xn
<= S  <=
B1*Y1  + B2*Y2 + .... + Bm*Ym 
as required.

Observation

If x'= (X1', X2', ... , Xn') is a primal feasible solution and y'= (Y1', Y2', ... , Ym') is a dual feasible solution and
C1*X1' + C2*X2' + .... + Cn*Xn'
= S =
B1*Y1' + B2*Y2' + .... + Bm*Ym'
then x' is an optimal solution to the primal and y' is an optimal solution to the dual.

Why?

Every primal feasible solution satisfies
C1*X1 + C2*X2 + .... + Cn*Xn <= S
= B1*Y1' + B2*Y2' + .... + Bm*Ym'
so the primal solution must be optimal.

Every dual feasible solution satisfies
C1*X1' + C2*X2' + .... + Cn*Xn' = S
<= B1*Y1 + B2*Y2 + .... + Bm*Ym
so the dual solution must be optimal.