Gruber P. Convex and Discrete Geometry

340 Convex Polytopes
Description of a Standard Version of the Simplex Algorithm, Given a Vertex
Let a1,...,am be the row vectors of A. The vertex v0 of the feasible set P =
{x : Ax ≤ b} is the intersection of a suitable subfamily of d hyperplanes among
the m hyperplanes Ax = b, sayA0x = b0, where A0 is a non-singular d × d submatrix
of A and b0 is obtained from b by deleting the entries corresponding to the
rows of A not in A0. Clearly, A0v0 = b0 and, since A0 is non-singular, we may
represent the row vector c as a linear combination of the rows of A0,say
(2) c = u0 A.
Here u0 is a row vector in Em with entries outside (the row indices of) A0 equal to 0.
We distinguish two cases:
(i) u0 ≥ o. Then c is a non-negative linear combination of the rows of A0. The
hyperplane {x : cx = cv0} thus supports P at v0 and has exterior normal vector c by
Proposition 20.1. Hence v0 is an optimum solution of (1).
(ii) u0 �≥ o. Leti0be the smallest index of a row in A0 with u0i0 < 0. Since A0
is a non-singular d × d matrix, the following system of linear equations has a unique
solution x0:
(3) ai x0 = 0 for each row ai of A0, except for ai0 for which ai0 x0 =−1.
Then v0 + λx0 is on a bounded edge of P, on an unbounded edge of P, or outside P
for λ>0. In addition,
(4) cx0 = u0 Ax0 =−u0i0 > 0.
Case (ii) splits into two subcases:
(iia) ax0 ≤ 0 for each row a of A. Then v0 + λx0 ∈ P for all λ ≥ 0. Noting (4),
it follows that the supremum in (1) is +∞.
(iib) ax0 > 0 for a suitable row a of A.Letλ0 ≥ 0 be the largest λ ≥ 0 such that
v0 + λx0 ∈ P, i.e.
λ0 = max � λ ≥ 0 : Av0 + λAx0 ≤ b �
= max � λ ≥ 0 : λa j x0 ≤ β j − a jv0 for all j = 1,...,m with a j x0 > 0 � .
Let j0 be the smallest index j of a row in A for which the maximum is attained.
Let A1 be the d × d matrix which is obtained from the non-singular d × d matrix A0
by deleting the row ai0 and inserting the row a j0 at the appropriate position. Since
a j0 x0 > 0 by our choice of j0, (3) and the fact that A0 is non-singular show that
A1 is also non-singular. Since ai(v0 + λ0x0) = βi for each row of the non-singular
matrix A1, by (3) and the choice of A0 and j0, and since ai(v0 + λ0x0) ≤ βi for
all other rows of A, by the definition of λ0, it follows that v1 = v0 + λ0x0 is a also
vertex of P.WehaveA1v1 = b1, where b1 is obtained from b0 by deleting the entry
βi0 and inserting the entry β j0 at the appropriate position (Fig. 20.1).
Repeat this step with v1, A1 instead of v0, A0.