Linear Programming Lecture Notes - Penn State Personal Web Server

if i ∈ I1. This confirms (since b ′
i = 0) row i of the augmented matrix [b|B −1 ] is
lexicographically positive.
In the second case that i ∈ I1, then we may proceed to determine whether i ∈ I2.
This process continues until we identify the j for which Ij is the singleton index r.
Such a j must exist by Lemma 7.5. In each case, we may reason that row i of the
augmented matrix [b|B −1 ] is lexicographically positive.
The preceding argument shows that at step n+1 of the simplex algorithm we will arrive an
augmented matrix [b|B −1 ] for which every row is lexicographically positive. This completes
the proof.
Remark 7.8. The assumption that we force Im into the basis can be justified in one of
two ways:
(1) We may assume that we first execute a Phase I simplex algorithm with artificial
variables. Then the forgoing argument applies.
(2) Assume we are provided with a crash basis B and we form the equivalent problem:
P ′
⎧
⎪⎨ max 0
⎪⎩
T xB + (c T N − c T BB −1 N)xN
s.t. ImxB + B −1 NxN = B −1 b
xB, xN ≥ 0
where B −1 b ≥ 0. This problem is clearly equivalent because its initial simplex
tableau will be identical to a simplex tableau generated by Problem P with basis
matrix B. If no such crash basis exists, then the problem is infeasible.
Lemma 7.9. Under the assumptions of Lemma 7.7, let zi and zi+1 be row vectors in
R n+1 corresponding to Row 0 from the simplex tableau at iterations i and i + 1 respectively.
Assume, however, that we exchange the z column (column 1) and the RHS column (column
n + 2). Then zi+1 − zi is lexicographically positive.
Proof. Consider the tableau in Table 1. If we are solving a maximization problem, then
clearly for xk to be an entering variable (as we assumed in the proof of Lemma 7.7) we must
have zk − ck < 0. Then the new Row Zero is obtained by adding:
y = −(zk − ck)
0 ar1 . . . arj . . . arm arm+1 . . . ark . . . arn br
ark
to the current row zero consisting of [1 z1 − c1 . . . zn − cn z]. That is: zi+1 = zi + y, or
y = zi+1 − zi.
The fact that zk − ck < 0 and ark > 0 (in order to pivot at that element) implies that
−(zk − ck)/ark > 0. Further, Lemma 7.7 asserts that the vector [0 ar1 . . . arn br] is
lexicographically positive (if we perform the exchange of column 1 and column n + 2 as we
assumed we would). Thus, y is lexicographically positive by Lemma 7.4. This completes the
proof.
Theorem 7.10. Under the assumptions of Lemma 7.7, the simplex algorithm converges
in a finite number of steps.
Proof. Assume by contradiction that we begin to cycle. Then there is a sequence of
row 0 vectors z0, z1, . . . , zl so that zl = z0. Consider yi = zi − zi−1. By Lemma 7.9, yi ≻ 0
115