Formula Sheet
•Inverse of a nonsingular 2 ×2 matrix.
a b
c d −1
=1
ad −bc d−b
−c a
•Given the LP
max cTx
Ax =b
x≥0.
•The primal dictionary in the matrix form is given by,
Z=cT
BB−1b−(cT
BB−1N−cT
N)xN
xB=B−1b−B−1NxN.
•The dual dictionary is the negative-transpose given by,
−W=−cT
BB−1b−(B−1b)TyB
yN= (B−1N)TcB−cN+ (B−1N)TyB.
•With the notation x∗
B=B−1b, y∗
N= (B−1N)TcB−cNand Z∗=cT
BB−1b=cT
Bx∗
B,
the dictionaries are
Z=Z∗−y∗T
NxN−W=−Z∗−x∗T
ByB
xB=x∗
B−B−1NxNyN=y∗
N+ (B−1N)TyB
•The Primal Simplex Algorithm in Matrix Notation:
Input:B,N,x∗
B≥0 , x∗
N= 0, y∗
Nand y∗
B= 0
while y∗
N0do
pick entering variable xjas j∈ N for which y∗
j<0;
i=argmin{(x∗
B)B(i)
(B−1aj)B(i): (B−1aj)B(i)>0, i ∈ B};
if no such iexists then
return LP is unbounded
else
θ=(x∗
B)B(i)
(B−1aj)B(i)and xileaves;
end
x∗N←θeN(j);
x∗
B←x∗
B−θB−1aj;
α=−y∗
j
[(B−1N)TeB(i)]N(j);
y∗
N←y∗
N+α(B−1N)TeB(i);
y∗
B←αeB(i);
B ← B ∪ {j}\{i};
N ← N ∪ {i}\{j};
end
