Stochastic Programming - Index of

58 STOCHASTIC PROGRAMMING
Figure 21
LP: bounded feasible set.
and |I(w)| ≤ k such that, according to our induction assumption, with
{x {i} , i =1, ···,r} the set of all feasible basic solutions, v = ∑ r
i=1 λ ix {i} ,
where ∑ r
i=1 λ i = 1, λ i ≥ 0 ∀i, andw = ∑ r
i=1 µ ix {i} ,where ∑ r
i=1 µ i =
1, µ i ≥ 0 ∀i. As is easily checked, we have ˆx = ρv +(1− ρ)w with
ρ = −β/(α − β) ∈ (0, 1). This implies immediately that ˆx is a convex linear
combination of {x {i} , i =1, ···,r}.
✷
The convex hull of finitely many points {x {1} , ···,x {r} }, formally denoted
by conv{x {1} , ···,x {r} }, is called a convex polyhedron or a bounded convex
polyhedral set (see Figure 21). Take for instance in IR 2 the points z 1 =
(2, 2),z 2 =(8, 1),z 3 =(4, 3),z 4 =(7, 7) and z 5 =(1, 6). In Figure 22 we
have ˜P = conv{z 1 , ···,z 5 }, and it is obvious that z 3 is not necessary to
generate ˜P; inotherwords, ˜P =conv{z 1 ,z 2 ,z 3 ,z 4 ,z 5 } =conv{z 1 ,z 2 ,z 4 ,z 5 }.
Hencewemaydropz 3 without any effect on the polyhedron ˜P, whereas
omitting any other of the five points would essentially change the shape of
the polyhedron. The points that really count in the definition of a convex
polyhedron are its vertices (z 1 ,z 2 ,z 4 and z 5 in the example). Whereas in twoor
three-dimensional spaces, we know by intuition what we mean by a vertex,
we need a formal definition for higher-dimensional cases: A vertex of a convex
polyhedron P is a point ˆx ∈P such that the line segment connecting any two
points in P, both different from ˆx, does not contain ˆx. Formally,
̸ ∃y, z ∈P,y≠ˆx ≠ z, λ ∈ (0, 1), such that ˆx = λy +(1− λ)z.
It may be easily shown that for an LP with a bounded feasible set B the
feasible basic solutions x {i} , i =1, ···,r, coincide with the vertices of B.
By Proposition 1.10, the feasible set of a linear program is a convex