Gruber P. Convex and Discrete Geometry

342 Convex Polytopes
(6) u piai xq > 0 for at least one row ai in A p.
We now distinguish several cases:
i < n: Then u pi ≥ 0 since n is the smallest index of a row in A p with
u pn < 0. In addition, ai xq ≤ 0 since n is the smallest index of a
row in A with anvq = βn and anxq > 0 (note that λq = 0).
i = n: Then u pn < 0 and anxq > 0, see the case i < n.
i > n: Then ai xq = 0 since ai is not deleted from Aq,see(3).
Each case is in contradiction to (6), concluding the proof of the theorem. ⊓⊔
What to do, if no Vertex is Known?
If no vertex of the feasible set P ={x : Ax ≤ b} �= ∅of a linear optimization
problem
(7) sup{cx : Ax ≤ b}
is known, we proceed as follows: Let S be the linear subspace lin{a1,...,am} of E d .
Then it is easy to see that
P = Q ⊕ S ⊥ ,
where Q = P ∩ S is a convex polyhedron with vertices. If c �∈ S, then cx assumes
arbitrarily large values on P = Q ⊕ S ⊥ . Then the supremum is +∞ and we are
done. If c ∈ S, then
sup{cx : x ∈ P} =sup{cx : x ∈ Q}.
Thus we have reduced our problem (7) to a problem where it is clear that the feasible
set has vertices, but we do not know them.
Changing notation, we assume that the new problem has the form
sup{cx : Ax ≤ b},
where the feasible set P ={x : Ax ≤ b} has vertices. Consider the following linear
optimization problem with one more variable z (Fig. 20.2),
(8) sup � z : Ax − bz ≤ o, −z ≤ 0, z ≤ 1 �
with the feasible set
Q = cl conv � {(o, 0)}∪ � P + (o, 1) �� .
(o, 0) is a vertex of Q. LetA0 be a non-singular d × d sub-matrix of A. Then (o, 0)
is the intersection of the d + 1 hyperplanes
ai x − βi z = 0 , ai row of A0,
z = 0.