Gruber P. Convex and Discrete Geometry

338 Convex Polytopes
Existence of Solutions
Since each convex polyhedron P �= ∅can be represented in the form
� λ1 p1 +···+λm pm : λi ≥ 0, λ1 +···+λm = 1 � + � µ1q1 +···+µnqn : µ j ≥ 0 � ,
the following result is easy to see:
Proposition 20.2. Assume that the linear optimization problem sup{cx : Ax ≤ b}
has non-empty feasible set P ={x : Ax ≤ b}. Then the supremum is either +∞,
or it is finite and attained. An analogous result holds for the problem inf{yb : y ≥
o, yA = c}.
The Duality Theorem
of von Neumann [768] and Gale, Kuhn and Tucker [353] is as follows.
Theorem 20.1. Let A be a real m × d matrix, b ∈ E m , c ∈ E d . If at least one of the
extreme sup{cx : Ax ≤ b} and inf{yb : y ≥ o, yA = c} is attained, then so is the
other and
(1) max{cx : Ax ≤ b} =min{yb : y ≥ o, yA = c}.
Proof. We first show the following inequality, where P = {x : Ax ≤ b} and
Q ={y : y ≥ o, yA = c}:
(2) If P, Q �= ∅, then sup{cx : Ax ≤ b} and inf{yb : y ≥ o, yA = c} both are
attained and
max{cx : Ax ≤ b} ≤min{yb : y ≥ o, yA = c}.
Let x ∈ P, y ∈ Q. Then cx = yAx ≤ yb. Hence sup{cx : Ax ≤ b} ≤inf{yb : y ≥
o, yA ≤ c} and both are attained by Proposition 20.2, concluding the proof of (2).
Assume now that P �= ∅and that δ = sup{cx : Ax ≤ b} is attained at p ∈ P =
{x : Ax ≤ b}, say. Then p is a boundary point of P and the hyperplane through p
with normal vector c supports P at p. Clearly, c is an exterior normal vector of this
support hyperplane. Let a1x ≤ β1,...,akx ≤ βk be the inequalities among the m
inequalities Ax ≤ b, which are satisfied by p with the equality sign. We may assume
that a1,...,ak are the first k row vectors of A and β1,...,βk the first k entries of b.
Proposition 20.1 then implies that
Thus,
(3) c = λ1a1 +···+λkak with suitable λi ≥ 0.
and therefore
δ = cp = λ1a1 p +···+λkak p = λ1β1 +···+λkβk
max{cx : Ax ≤ b} =δ = cp = λ1β1 +···+λkβk
= (λ1,...,λk, 0,...,0)b ≥ inf{yb : y ≥ o, yA = c},