Gruber P. Convex and Discrete Geometry

20 Linear Optimization 339
on noting that (λ1,...,λk, 0,...,0) ∈ Q = {y : y ≥ o, yA = c} by (3). In
particular, Q �= ∅. (2) then shows that the infimum is also attained and that the
maximum is less than or equal to the minimum. Thus (1) holds in case where P �= ∅
and the supremum is attained.
Assume, finally, that Q �= ∅and that ε = inf{yb : y ≥ o, yA = c} is attained at
q ∈ Q ={y : yA = c, −y ≤ o}, say. Then −b is an exterior normal vector of the
support hyperplane {y : yb = ε} of Q at q. Proposition 20.1 then shows that
(4) b = µ1b1 +···+µdbd + ν1e1 +···+νlel with suitable µi ∈ R,νj ≥ 0,
where b1,...,bd are the column vectors of A and e1,...,el are standard unit vectors
of E d such that e j has entry 1 where q has entry 0. In particular,
(5) qej = 0for j = 1,...,l.
(4) implies that
A(µ1,...,µd) T = µ1b1 +···+µdbd = b − ν1e1 −···−νlel ≤ b.
Thus (µ1,...,µd) T ∈ P and therefore P �= ∅. This, together with (4) and (5),
shows that
min{yb : y ≥ o, yA = c} =ε = qb = µ1qb1 +···+µdqbd
= qA(µ1,...,µd) T = c(µ1,...,µd) T
≤ sup{cx : Ax ≤ b}.
Since P, Q �= ∅, (2) shows that the supremum is also attained and that the maximum
is less than or equal to the minimum. Thus (1) holds also in the case where Q �= ∅
and the infimum is attained. The proof of the theorem is complete. ⊓⊔
20.2 The Simplex Algorithm
The simplex algorithm of Dantzig [237], or versions of it, are still the common
method for linear optimization problems.
In this section we give a description of the simplex algorithm and show that it
leads to a solution. The presentation follows Schrijver [915].
The Idea of the Simplex Algorithm
Consider a linear optimization problem of the form
(1) sup{cx : Ax ≤ b}
where a vertex v0 of the feasible set P ={x : Ax ≤ b} is given. Check whether there
is an edge of P starting at v0 along which the objective function cx increases. If there
is no such edge, v0 is an optimum solution. Otherwise move along one of these edges.
If this edge is a ray, the supremum is infinite. If not, let v1 be the other vertex on this
edge. Repeat this step with v1 instead of v0, etc. This leads to an optimum solution
or shows that the supremum is infinite in finitely many steps.